The base, the exponent and the power; reading and writing powers in words and in exponential form; evaluating powers including powers of a variable; how grouping symbols decide what an exponent is attached to; and the area and volume formulas that give the second and third powers their names.
Subject: Algebra 1 · 65 slides · symbolic lesson
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Title
Algebra 1 · Chapter 1 — Connections to Algebra
Exponents and Powers
Objectives
Five outcomes, each one you can test yourself on with a pencil and no answer key.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.2 Exponents and Powers §1.2, pp. 9-14 — the lesson these objectives are drawn from
Warm-up
The notation is new. The arithmetic underneath it is not.
Discussion prompt
Work out two times two times two times two, without writing the expression down. Then say how many twos you multiplied, and how you kept count.
Hint: Most people count the twos rather than the multiplication signs. That instinct is exactly what the new notation records.
Answer:
\[ 2 \cdot 2 \cdot 2 \cdot 2 = 16 \]
Four twos, and you almost certainly counted the twos rather than the three multiplication signs between them. Exponent notation writes down that count and nothing else: the small number records how many copies of the base are being multiplied. Everything in this lesson follows from that one sentence.
Concept
An expression such as two to the third power is called a power. The small raised number is the exponent, and it records how many times the base is used as a factor. It is a counter, not a multiplier.
power — An expression consisting of a base and an exponent, such as two to the third power. The exponent tells how many times the base is used as a factor.
\[ a^n = \underbrace{a \cdot a \cdot a \cdots a}_{n \text{ times}} \]
Nearly every mistake in this lesson comes from reading the exponent as a multiplier instead of a counter.
Figure (svg): The power 2 to the third labelled with base, exponent and power, expanded as 2 times 2 times 2 with the three factors marked
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.2 Exponents and Powers §1.2, pp. 9-9
Section
Section 1
Concept
The base is the number being multiplied. The exponent is the small raised number that counts the copies. The power is the whole expression, base and exponent together.
exponent — The number that tells how many times the base is used as a factor. In two to the third power, the exponent is three.
Figure (svg): The power 2 to the third labelled with base, exponent and power, expanded as 2 times 2 times 2 with the three factors marked
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.2 Exponents and Powers §1.2, pp. 9-9 — the base-and-exponent diagram
Picture it
Watch what changes from one row to the next. It is not the amount added.
Figure (svg): Four rows showing 2 to the first, second, third and fourth powers written out as repeated factors and evaluated
Each row has one more factor of two than the row above, so each answer is double the one above rather than two more than it. Confusing those two readings is the whole misconception this lesson has to remove.
Worked example
Trivial once you have the vocabulary, and worth being certain about before anything harder.
\[ \text{In } 2^3, \text{ name the base, the exponent and the power, and write out its meaning.} \]
The base is 2
Why: The base is the number on the line, the one actually being multiplied.
\[ \text{base } = 2 \]
The exponent is 3
Why: The exponent is the raised number, and it counts factors.
\[ \text{exponent } = 3 \]
The power is the whole expression
Why: Power names the object, not one of its two pieces.
\[ \text{power } = 2\text{ to the third} \]
Write the meaning as repeated factors
Why: Three factors of two, joined by multiplication signs.
\[ 2 \cdot 2 \cdot 2 \]
Evaluate
Why: Two times two is four, times two again is eight.
\[ 8 \]
Figure (svg): The solution to Worked example name every part shown as a ladder of expressions, one row per algebraic move
\[ 2^3 = 2 \cdot 2 \cdot 2 = 8 \]
Verify: count the factors against the exponent
Why: There are exactly three twos written out, matching the exponent of three. If the count of factors ever disagrees with the exponent, the expansion is wrong before the arithmetic even starts.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.2 Exponents and Powers §1.2, pp. 9-9
Sorting
In each expression below, decide what role the number 3 is playing.
Sort into buckets
Drop each expression into the column that says what the 3 is doing.
The first two items use the same two digits and mean completely different numbers: nine and eight. Position is meaning here, which is not true of any notation you met before this lesson.
Worked example
Three powers of the same base. Write each one out in full before touching a calculator.
\[ \text{Write out and evaluate } \; 3^2, \quad 3^3, \quad 3^4. \]
Three squared is two factors of three
Why: The exponent two means two copies of the base.
\[ 3 \cdot 3 = 9 \]
Three cubed is three factors of three
Why: One more copy than the line above, so the previous answer times three.
\[ 9 \cdot 3 = 27 \]
Three to the fourth is four factors of three
Why: Again one more copy, so the previous answer times three.
\[ 27 \cdot 3 = 81 \]
Notice the pattern between the answers
Why: Each answer is three times the one above, because each has one extra factor of three.
\[ 9, 27, 81 \]
Figure (svg): The solution to Worked example expand before you multiply shown as a ladder of expressions, one row per algebraic move
\[ 3^2 = 9, \quad 3^3 = 27, \quad 3^4 = 81 \]
Verify: divide each answer by the one above it
Why: Twenty-seven divided by nine is three, and eighty-one divided by twenty-seven is three. Every step up the list multiplies by exactly one more base, which is what the exponent claimed.
Trap
\[ 2^3 \]
Read it as two times three
Why: The two numbers are written right next to each other, and in Lesson 1.1 that meant multiply.
\[ 2^3 = 6 \quad \text{(wrong)} \]
This gets the right answer for exactly one case, two to the second power, which is why it can survive undetected for a surprisingly long time.
\[ 2^3 = 2 \cdot 2 \cdot 2 = 8 \]
Read the exponent as a count of factors, not as a factor itself
Why: The raised position is doing real work: it takes the number out of the multiplication and turns it into an instruction about the multiplication.
\[ 2^2 = 4 = 2 \cdot 2 \quad \text{but} \quad 2^3 = 8 \neq 2 \cdot 3 \]
Test any suspected case at the third power rather than the second, because the second power is where the wrong rule and the right rule happen to agree.
Elimination
All four claims are about the expression five cubed. Only one of them is true.
Eliminate the wrong options
Which statement about five cubed is correct?
Survives elimination: B
Why: Five cubed is three factors of five: five times five is twenty-five, times five again is one hundred and twenty-five. The three wrong options are the three standard misreadings — treating the exponent as a multiplier, swapping base and exponent, and confusing repeated multiplication with repeated addition.
Notation
This line is the definition the whole lesson rests on. Take it apart piece by piece.
Annotate
On: \( a^n = \underbrace{a \cdot a \cdot a \cdots a}_{n \text{ times}} \)
Read the whole line as a sentence: a to the n means a used as a factor n times. If you can say that sentence, you can reconstruct the definition without looking at it.
Prediction
Commit before you compute. Most people underestimate this badly.
Predict first
Two to the third power is 8. What is two to the tenth power?
Correct: Between 1000 and 1100 — it is 1024.
\[ 2^3 = 8 \quad 2^5 = 32 \quad 2^8 = 256 \quad 2^{10} = 1024 \]
Why: Each step up the powers of two doubles the previous answer, so going from the third power to the tenth doubles seven more times. Eight doubled seven times is 16, 32, 64, 128, 256, 512, 1024. Repeated multiplication outruns repeated addition very quickly, and that gap is the entire subject of Chapter 8.
Section
Section 2
Concept
The second power is read squared and the third power is read cubed, because those are the powers geometry uses for area and volume. Every other power is read as to the nth power.
Figure (svg): A three-column table pairing exponential form with the words for it and its meaning as repeated factors
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.2 Exponents and Powers §1.2, pp. 9-9 — Example 1, Read and Write Powers
Picture it
Cover any one column and reconstruct it from the other two. Do that in all three directions.
Figure (svg): A three-column table pairing exponential form with the words for it and its meaning as repeated factors
Going from words to symbols is the direction that appears on tests, because it is the direction a word problem forces on you.
Worked example
This is Example 1 from the textbook, all three parts.
\[ \text{Express } 4^2, \; 5^3, \; x^6 \text{ in words, then write each meaning.} \]
Four to the second power, or four squared
Why: The second power has a geometric nickname because two copies of a length fill a square.
\[ 4 \cdot 4 \]
Five to the third power, or five cubed
Why: The third power has a nickname for the same reason, one dimension further up.
\[ 5 \cdot 5 \cdot 5 \]
x to the sixth power
Why: Past the third power there is no nickname, so the ordinal is spoken in full.
\[ x \cdot x \cdot x \cdot x \cdot x \cdot x \]
Count the factors in each meaning against its exponent
Why: Two, three and six factors respectively — the count is the whole content of the exponent.
\[ 2, 3\text{ and } 6\text{ factors} \]
Figure (svg): The solution to Worked example express each power in words and in meaning shown as a ladder of expressions, one row per algebraic move
\[ 4^2 = 4 \cdot 4, \quad 5^3 = 5 \cdot 5 \cdot 5, \quad x^6 = x \cdot x \cdot x \cdot x \cdot x \cdot x \]
Verify: read each expansion back and count aloud
Why: Counting the factors in each expansion gives two, three and six, exactly matching the three exponents. The count is the only thing that can go wrong in this translation, so it is the only thing worth checking.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.2 Exponents and Powers §1.2, pp. 9-9
Translation
Four phrases, four expressions. Two of them use nicknames and two spell the ordinal out.
Match the pairs
Why: In every case the word naming the power becomes the exponent and everything else becomes the base. Squared means the exponent two and cubed means the exponent three, so those two phrases hide their exponent inside a single word — which is exactly why they are the two that get written backwards most often.
Worked example
Guided Practice 1 to 3, the direction that matters most. Write each one before advancing.
\[ \text{Write in exponential form: three squared, } \; x \text{ to the fourth power, } \; s \text{ cubed.} \]
Three squared has base 3 and exponent 2
Why: Squared always means the second power, whatever the base.
\[ 3 ^{2} \]
x to the fourth power has base x and exponent 4
Why: The base is whatever is being multiplied; here it is a variable rather than a number, which changes nothing.
\[ x ^{4} \]
s cubed has base s and exponent 3
Why: Cubed always means the third power.
\[ s ^{3} \]
Check that no base and exponent got swapped
Why: Reading each answer back in words is the fastest check: three squared, not two to the third.
\[ 3 ^{2}, x ^{4}, s ^{3} \]
Figure (svg): The solution to Worked example from words back to symbols shown as a ladder of expressions, one row per algebraic move
\[ 3^2, \quad x^4, \quad s^3 \]
Verify: read every answer back into words
Why: Reading them back gives three squared, x to the fourth power and s cubed — the three phrases we started from. A swap of base and exponent would be obvious the moment it was read aloud, which is why reading back is worth the four seconds.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.2 Exponents and Powers §1.2, pp. 9-9
Trap
\[ \text{three squared} \;\rightarrow\; 2^3 \]
Write the numbers in the order they were spoken
Why: The word three came first, so the digit 3 gets written first, and the raised position is filled with whatever is left.
\[ 2^3 = 8 \quad \text{but three squared is } 9 \]
The two answers are close enough that the error rarely announces itself, and far enough apart to lose the mark.
\[ \text{three squared} \;\rightarrow\; 3^2 \]
Identify the roles before writing anything
Why: The word squared names the exponent, so the exponent is two. Everything else in the phrase is the base.
\[ 3^2 = 3 \cdot 3 = 9 \]
The reliable habit is to ask which word tells you the exponent. In three squared it is squared; in x to the fourth power it is fourth. Whatever is left over is the base.
Matching
Four powers, four values. Two of the values are deliberately close together.
Match the pairs
Why: Two to the third is eight and three to the second is nine, so swapping base and exponent changes the answer even when the digits do not change. The last two are a genuine coincidence — two to the fourth and four to the second really are both sixteen — and coincidences like that are exactly why you should never conclude that swapping is safe.
Two truths and a lie
Look carefully at the position of every digit before you decide.
Eliminate the wrong options
Which of these is NOT equal to sixteen?
Survives elimination: D
Why: Four cubed is three factors of four: sixteen times four again, which is sixty-four rather than sixteen. The first three really do all equal sixteen, which makes this a useful reminder that two different powers can land on the same value without the swap being legal in general.
Faded example
The first two steps are done for you. Supply the count and the value.
Fill in the blanks
x^4 \text81 x = 3 \;=\; 3 \cdot 3 \cdot 3 \cdot 3 \;=\; 4 \qquad \text___ = ___
Why: Four factors of three multiply to eighty-one, and the number of factors is simply the exponent read off the original expression. The two blanks are deliberately paired: the count is the thing you can always state instantly, and it is the check on whether the expansion you multiplied was the right length.
Section
Section 3
Concept
Evaluating a power uses the same routine as Lesson 1.1, with one extra line inserted: after substituting, write the factors out in full before you multiply any of them.
The middle line is what stops the exponent from being quietly treated as a multiplier.
Figure (svg): Four rows showing 2 to the first, second, third and fourth powers written out as repeated factors and evaluated
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.2 Exponents and Powers §1.2, pp. 10-10 — Example 2, Evaluate the Power
Picture it
Four rows, each one factor longer than the one above it.
Figure (svg): Four rows showing 2 to the first, second, third and fourth powers written out as repeated factors and evaluated
Once the factors are written out there is no exponent left to misread — the line in front of you is ordinary multiplication, and ordinary multiplication is something you already do reliably.
Worked example
This is Example 2 from the textbook. Three lines, and the middle one is the point.
\[ \text{Evaluate } x^4 \text{ when } x = 2. \]
Substitute 2 for x
Why: The base was a variable and is now a number; the exponent never changes during a substitution.
\[ x ^{4} = 2 ^{4} \]
Write out the factors
Why: Four copies of the base, because the exponent is four.
\[ 2 \cdot 2 \cdot 2 \cdot 2 \]
Multiply the factors
Why: Two times two is four, times two is eight, times two is sixteen.
\[ 16 \]
State the value
Why: The value of the power is a single number.
\[ 16 \]
Figure (svg): The solution to Worked example evaluate x to the fourth at x equal to 2 shown as a ladder of expressions, one row per algebraic move
\[ x^4 = 2^4 = 2 \cdot 2 \cdot 2 \cdot 2 = 16 \]
Verify: count the twos you actually wrote
Why: Four twos are written out, matching the exponent of four. If you had written three the answer would have been eight, and counting is a far faster check than redoing the multiplication.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.2 Exponents and Powers §1.2, pp. 10-10
Faded example
The substitution and the expansion are done. Supply the value.
Fill in the blanks
s^3 \text8 s = 2 \;=\; 2^3 \;=\; 2 \cdot 2 \cdot 2 \;=\; ___
Why: Once the factors are written out there is no algebra left, only multiplication: two times two is four, and four times two is eight. Writing the expansion as its own line is what turns an exponent question into an arithmetic question, and arithmetic is the part you can already do without thinking.
Worked example
Example 3 part a from the textbook, with a equal to 1 and b equal to 2. Each power is evaluated separately, then combined.
\[ \text{Evaluate } \; (a^2) + (b^2) \; \text{ when } a = 1 \text{ and } b = 2. \]
Substitute 1 for a and 2 for b
Why: Both substitutions happen before any arithmetic, and the brackets keep the two powers separate.
\[ (1 ^{2}) + (2 ^{2}) \]
Write out the factors inside each bracket
Why: Two factors in each, because both exponents are two.
\[ (1 \cdot 1) + (2 \cdot 2) \]
Multiply inside each bracket
Why: The brackets are the innermost grouping, so they are finished first.
\[ 1 + 4 \]
Add
Why: Only now, with both powers reduced to numbers, does the addition happen.
\[ 5 \]
Figure (svg): The solution to Worked example two powers added together shown as a ladder of expressions, one row per algebraic move
\[ (a^2) + (b^2) = (1^2) + (2^2) = 1 + 4 = 5 \]
Verify: check each power separately against its own size
Why: One squared has to be one because multiplying ones can never leave one, and two squared is four. Adding those gives five, and no step mixed the two bases together — which is the mistake the next example is built to expose.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.2 Exponents and Powers §1.2, pp. 10-10
Error analysis
The student was asked to evaluate x to the fourth at x equal to 2, and then two cubed. Both answers are wrong, for two different reasons.
Annotate
On: \( x^4 = 2^4 = 2 \cdot 4 = 8 \qquad 2^3 = 2 + 2 + 2 = 6 \)
Notice that at the second power both wrong methods happen to give the right answer. Always test a suspected rule at the third power or higher.
Ranking
Four moves, one correct order. Getting the middle two the wrong way round is the error the whole section is about.
Put in order
Why: Substituting first turns the base from a letter into a number. Expanding second makes the exponent disappear into an ordinary product. Multiplying third finishes the arithmetic, and counting last is the check that the expansion was the right length. Multiplying before expanding is exactly how an exponent gets treated as a factor.
Prediction
Both of these use the digits two and five. Commit before you compute.
Predict first
Which is larger, two to the fifth power or five to the second power?
Correct: Two to the fifth, which is 32.
\[ 2^5 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 32 \qquad 5^2 = 5 \cdot 5 = 25 \]
Why: Two to the fifth is five factors of two, giving 2, 4, 8, 16, 32. Five to the second is two factors of five, giving 25. Swapping the base and the exponent gives a different number here, as it does almost always — the coincidences where it does not, such as two to the fourth and four to the second, are rare enough to be traps rather than rules.
Socratic
For small powers you can often see the answer. That is exactly why this question is worth answering now.
Discussion prompt
Give one reason to write the factors out even when the multiplication is easy, and describe one specific wrong answer that the expansion line would have caught.
Hint: Think about what the expansion makes countable that the original expression does not.
Answer:
The expansion turns the exponent into something you can count. In the original expression the exponent is a claim you have to remember to interpret; in the expansion it is a number of items on the page, and miscounting three twos as two twos is visible in a way that misreading an exponent is not.
Concretely, the expansion catches the answer 6 for two cubed. Written as two plus two plus two the error is obvious, because there are plus signs where multiplication signs belong. Written as two to the third with no working, six looks like a perfectly reasonable number.
Section
Section 4
Concept
Parentheses and brackets are grouping symbols. They tell you the order in which to do operations, and you always work the innermost group first. When an exponent is involved they also decide what the exponent is attached to.
grouping symbols — Parentheses and brackets, which tell you the order in which to carry out operations. The operations inside the innermost pair are done first.
\[ 2x^3 = 2(x^3) \qquad \text{but} \qquad (2x)^3 = (2x)(2x)(2x) \]
With no brackets, the exponent attaches only to the symbol immediately before it.
Figure (svg): Two panels comparing 2 x cubed with the quantity 2 x all cubed, evaluated at x equal to 4, giving 128 and 512
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.2 Exponents and Powers §1.2, pp. 10-11 — the Grouping Symbols box and Example 4
Picture it
Same digits, same value of x, and a difference of nearly four hundred.
Figure (svg): Two panels comparing 2 x cubed with the quantity 2 x all cubed, evaluated at x equal to 4, giving 128 and 512
On the left the exponent takes only the x, so the 2 is multiplied on at the end. On the right the exponent takes the whole product, so the 2 gets cubed as well. Nothing but the brackets caused that.
Worked example
This is Example 4 from the textbook, both parts, at x equal to 4.
\[ \text{Evaluate at } x = 4: \quad \text{(a) } 2x^3 \quad \text{(b) } (2x)^3. \]
In part a, the exponent applies to x alone
Why: There are no brackets, so the exponent takes only the symbol directly before it. The expression means two times the quantity x cubed.
\[ 2 x ^{3} = 2(4 ^{3}) \]
Evaluate the power first, then multiply
Why: The power is the innermost operation once the substitution is made.
\[ 2(64) = 128 \]
In part b, the brackets put the exponent on the whole product
Why: The bracket is the innermost grouping, so it is completed before the exponent acts.
\[ (2 x) ^{3} = (2 \cdot 4) ^{3} \]
Multiply inside the bracket, then evaluate the power
Why: Two times four is eight, and eight cubed is five hundred and twelve.
\[ 8 ^{3} = 512 \]
Figure (svg): The solution to Worked example 2x cubed against the quantity 2x cubed shown as a ladder of expressions, one row per algebraic move
\[ 2x^3 = 128 \qquad (2x)^3 = 512 \]
Verify: compare the two answers by a factor
Why: Five hundred and twelve divided by one hundred and twenty-eight is four. That is exactly what you should expect: the bracketed version cubes the 2 as well, contributing two to the third rather than two to the first, and eight divided by two is four.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.2 Exponents and Powers §1.2, pp. 11-11
Discrimination
For each expression, decide whether the exponent applies to the variable alone or to the whole product.
Sort into buckets
Sort each expression by what its exponent is attached to.
Worked example
Example 3 part b, with a equal to 1 and b equal to 2. The bracket contains an addition this time.
\[ \text{Evaluate } \; (a + b)^2 \; \text{ when } a = 1 \text{ and } b = 2. \]
Substitute 1 for a and 2 for b
Why: The bracket stays in place through the substitution; it is part of the expression, not scaffolding.
\[ (1 + 2) ^{2} \]
Add within the parentheses
Why: The innermost grouping is done first, before the exponent is allowed to act on anything.
\[ (3) ^{2} \]
Write out the factors
Why: Two copies of the three that the bracket produced.
\[ 3 \cdot 3 \]
Multiply
Why: Three times three is nine.
\[ 9 \]
Figure (svg): The solution to Worked example brackets around a sum shown as a ladder of expressions, one row per algebraic move
\[ (a + b)^2 = (1 + 2)^2 = 3^2 = 9 \]
Verify: compare against the earlier example on the same numbers
Why: The earlier example gave one squared plus two squared, which was 5. This gives 9. Squaring a sum is not the same as adding the squares, and the two examples on identical numbers are the proof — a fact Chapter 10 will return to at length.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.2 Exponents and Powers §1.2, pp. 10-10
Trap
\[ (a + b)^2 \text{ at } a = 1, \; b = 2 \]
Square each term inside the bracket and add the results
Why: The exponent looks like it should share itself out over the plus sign the way a multiplier does.
\[ 1^2 + 2^2 = 1 + 4 = 5 \quad \text{(wrong)} \]
This is the single most persistent error in all of algebra, and it survives all the way into Chapter 10 unless it is killed here.
\[ (a + b)^2 \text{ at } a = 1, \; b = 2 \]
Finish the innermost grouping first, then apply the exponent
Why: The bracket is an instruction: add before you do anything else. The exponent acts on the single number the bracket produced.
\[ (1 + 2)^2 = 3^2 = 9 \]
Nine against five, on the smallest numbers available. Keep this pair — it is the cheapest counterexample you will ever carry, and it settles the question every time it comes up.
Elimination
All four expressions use the number 2 and the number 4 in some arrangement.
Eliminate the wrong options
Which expression equals 512?
Survives elimination: B
Why: The bracket is completed first, giving eight, and eight cubed is five hundred and twelve. The three wrong options each cube a different part of the expression, and each lands on a completely different answer — which is a compact demonstration that in an expression with an exponent, what the exponent is attached to matters more than which digits are present.
Comparison
Fill the blanks from memory before you scroll back.
Comparison matrix
| Expression at x equal to 4 | What the exponent takes | Value |
|---|---|---|
| 2x^3 | the letter x only | 128 |
| (2x)^3 | the whole product 2x | 512 |
The rule with no brackets present is worth stating in one sentence: an exponent attaches to the single symbol immediately in front of it, and to nothing else.
Edge cases
The two expressions in this section usually disagree. Find the values where they cannot.
Discussion prompt
For which values of x do 2x cubed and the quantity 2x all cubed give the same answer? Find them, and then say what is special about those numbers that makes the brackets stop mattering.
Hint: Try x equal to 0 first, and then think about what number leaves everything unchanged when it is a factor.
Answer:
\[ \text{At } x = 0: \quad 2(0^3) = 0 \quad \text{and} \quad (2 \cdot 0)^3 = 0 \]
\[ \text{At } x = 4: \quad 128 \neq 512 \]
Zero is the only value where the two agree. At zero both expressions collapse to zero, because a zero factor destroys everything regardless of where the brackets are. At every other value the bracketed version is eight times larger, since it contributes two cubed rather than two.
Testing at zero is fast, but this is a case where it is misleading on its own — an agreement at zero proves nothing about the general case. Always test a second, more awkward value.
Section
Section 5
Concept
A square of side s has area s squared, and a cube of edge s has volume s cubed. The nicknames for the second and third powers are borrowed directly from those two shapes, and so are the units.
\[ A = s^2 \qquad V = s^3 \]
Units follow the same rule: an area in square feet and a volume in cubic feet are written with a second and a third power of the unit.
Figure (svg): A square of side s beside a cube of edge s, labelled with the area formula s squared and the volume formula s cubed
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.2 Exponents and Powers §1.2, pp. 11-11
Picture it
The picture explains the vocabulary, and the vocabulary then explains the units.
Figure (svg): A square of side s beside a cube of edge s, labelled with the area formula s squared and the volume formula s cubed
A length is measured in feet, an area in square feet and a volume in cubic feet. The exponent on the number and the exponent on the unit always match, which makes the unit a free check on the formula.
Worked example
Example 5 from the textbook. The tank is a cube with an inner edge of two feet.
\[ \text{A cube-shaped tank has inner edge } s = 2 \text{ feet. Find its volume.} \]
Write the formula for the volume of a cube
Why: Starting from the general rule keeps the substitution visible.
\[ V = s ^{3} \]
Substitute 2 for s
Why: The edge length is two feet, and it goes in wherever the letter s appears.
\[ V = 2 ^{3} \]
Evaluate the power
Why: Three factors of two: two times two is four, times two is eight.
\[ V = 8 \]
Attach the unit
Why: Three lengths in feet were multiplied, so the unit carries a third power too.
\[ V = 8\text{ cubic feet} \]
Figure (svg): A cube-shaped fish tank with each edge labelled 2 feet, and the volume calculation 2 cubed equals 8 cubic feet beside it
\[ V = s^3 = 2^3 = 8 \text{ cubic feet} \]
Verify: check the unit against the exponent
Why: Three factors of a length in feet must give feet to the third power, which is read cubic feet. If the answer had come out in square feet the formula used would have been the wrong one, and the unit says so before the number does.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.2 Exponents and Powers §1.2, pp. 11-11
Matching
Four quantities, four units. The exponent decides every one.
Match the pairs
Why: An edge is a single length, so it carries the plain unit. An area comes from two lengths multiplied, so it carries the second power. A volume comes from three, so it carries the third. Adding lengths together, as in the total edge length, never changes the unit at all — only multiplying lengths does, which is the distinction most worth taking away.
Worked example
Guided Practice 10, using the same tank. One dimension fewer, and everything about the answer changes.
\[ \text{Find the area of one square face of the same tank, where } s = 2 \text{ feet.} \]
Write the formula for the area of a square
Why: A face of a cube is a square, so the area formula applies to it directly.
\[ A = s ^{2} \]
Substitute 2 for s
Why: The same edge length, used in a formula with a different exponent.
\[ A = 2 ^{2} \]
Evaluate the power
Why: Two factors of two.
\[ A = 4 \]
Attach the unit
Why: Two lengths in feet were multiplied, so the unit carries a second power.
\[ A = 4\text{ square feet} \]
Figure (svg): The solution to Worked example the area of one face shown as a ladder of expressions, one row per algebraic move
\[ A = s^2 = 2^2 = 4 \text{ square feet} \]
Verify: check the area against the volume
Why: The cube has six faces of four square feet each, and its volume is eight cubic feet. Those are different quantities in different units, so they cannot be compared as numbers — which is precisely why the unit has to be written down rather than assumed.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.2 Exponents and Powers §1.2, pp. 11-11
Trap
\[ V = s^3 = 2^3 = 8 \text{ square feet} \]
Compute the volume correctly, then attach whichever unit comes to mind
Why: The arithmetic felt like the hard part, so the unit gets treated as a label rather than as part of the answer.
Eight square feet describes a flat region, not a tank. The number is right and the answer is still wrong.
\[ V = s^3 = 2^3 = 8 \text{ cubic feet} \]
Read the exponent in the formula and let it choose the unit
Why: Three factors of a length give a third power of the unit, which is what cubic means.
\[ A = s^2 \rightarrow \text{ square feet} \qquad V = s^3 \rightarrow \text{ cubic feet} \]
The exponent on the number and the exponent on the unit always match. That match is a free check: if they disagree, the wrong formula was used.
Estimation
A cube-shaped box has an edge of about three feet.
Predict first
Roughly what volume does it hold?
Correct: About 27 cubic feet.
\[ V = s^3 = 3^3 = 27 \text{ cubic feet} \qquad A = s^2 = 9 \text{ square feet} \]
Why: Three cubed is three factors of three: nine, then twenty-seven. Nine would be the area of a face rather than the volume, twelve is the total edge length of a face, and six comes from multiplying the edge by the number of dimensions — all three wrong answers correspond to a real quantity, just not the one that was asked for.
Socratic
The exponent on the unit is not a convention someone chose. It follows from the multiplication.
Discussion prompt
Explain, using the fish tank, why multiplying two feet by two feet by two feet gives cubic feet rather than feet. Then say what unit you would get by multiplying an area in square feet by a length in feet, and why.
Hint: Treat the word feet as though it were a variable being multiplied along with the number.
Answer:
The units multiply alongside the numbers. Two feet times two feet times two feet is eight, and feet times feet times feet — three factors of the same unit, which is exactly what a third power records. Writing it as cubic feet is just saying feet to the third power in English.
Square feet times feet gives cubic feet, because two factors of feet multiplied by one more factor of feet makes three. This is why the units check works at all: they obey the same exponent rules the numbers do, so an impossible unit is a reliable sign of an impossible calculation.
Missing information
A question can be perfectly well written and still be unanswerable.
Discussion prompt
A tank has a volume of 27 cubic feet. What is the area of one face? Say exactly what assumption you have to make before this question has a single answer, and why it is not automatically true.
Hint: How many different shapes have a volume of 27 cubic feet?
Answer:
You have to assume the tank is a cube. A tank measuring 27 feet by 1 foot by 1 foot also has a volume of 27 cubic feet, and its faces are nothing like square. Volume alone does not determine shape.
\[ \text{If it is a cube: } s^3 = 27 \Rightarrow s = 3, \text{ so } A = 3^2 = 9 \text{ square feet} \]
Working backwards from a volume to an edge is a question Chapter 12 answers properly with cube roots. For now the point is the assumption, not the technique.
Comparison
Fill the blanks from memory. Every row is a real answer a student has written for two to the third power.
Comparison matrix
| Wrong reading | What it computes | Correct value |
|---|---|---|
| exponent as a multiplier | 2 times 3, giving 6 | 8 |
| exponent as repeated addition | 2 + 2 + 2, giving 6 | 8 |
| base and exponent swapped | 3 to the second, giving 9 | 8 |
Two of the three wrong readings happen to agree with each other, which is why a student can arrive at six twice by different routes and feel confirmed. Only writing the factors out separates them.
Pattern
Whether the question says evaluate, write in exponential form, or find a volume, the same five moves cover it.
Step three is what separates two x cubed from the quantity two x all cubed, and step four is what stops an exponent being read as a multiplier. Those are the two errors this lesson exists to prevent.
OpenStax Elementary Algebra 2e, §1.2 Use the Language of Algebra §1.2
Check
Reading the notation. Solve it before you click.
Check your understanding
What is the value of four squared?
Answer: B
Why: Squared means the second power, so four squared is two factors of four: four times four, which is sixteen. The exponent counts the copies of the base rather than acting as a factor itself.
Check
Grouping symbols. Write the substitution line before you choose.
Check your understanding
Evaluate the expression (3x) squared when x equals 2.
Answer: A
Why: The bracket is the innermost grouping, so it is finished first: three times two is six. Squaring six gives thirty-six. Because the bracket wraps the whole product, the 3 is squared along with the x.
Check
Formulas and units. Decide the unit before you compute the number.
Check your understanding
A cube-shaped storage box has an inner edge of 5 inches. What is its volume?
Answer: A
Why: Volume of a cube is the edge cubed, so five cubed is five times five times five, which is one hundred and twenty-five. Three factors of a length in inches give inches to the third power, read as cubic inches.
Real world
A digital photo is 1024 pixels wide and 1024 pixels tall. Someone tells you that doubling both dimensions doubles the file size.
Discussion prompt
Decide whether that claim is right, and say by what factor the number of pixels actually changes when both the width and the height are doubled. Then explain which of the two ideas from this lesson — the exponent as a counter, or the exponent on the unit — makes the answer obvious.
Hint: Count the pixels as an area rather than as two separate lengths.
Answer:
\[ \text{original: } 1024 \cdot 1024 \qquad \text{doubled: } 2048 \cdot 2048 = 4 \cdot (1024 \cdot 1024) \]
The claim is wrong: the number of pixels goes up by a factor of four, not two. Doubling a length doubles it once, but a pixel count is an area, and an area involves two factors of the length — so each factor doubles, and two doublings make four.
The unit is what makes this obvious. Pixels are counted per square, so the quantity carries a second power, and anything that scales a length by a factor scales a second power by that factor squared. The same reasoning says that doubling the edge of a cube multiplies its volume by eight.
Commit first
Answer, then rate your confidence honestly. Confident and wrong is the combination worth finding.
Predict first
Is the square of a sum always equal to the sum of the squares?
Correct: No, and it fails for almost every pair of numbers.
\[ (1 + 2)^2 = 3^2 = 9 \qquad 1^2 + 2^2 = 1 + 4 = 5 \]
\[ (a + b)^2 = a^2 + 2ab + b^2 \quad \text{— the missing piece is the middle term} \]
Why: One and two settle it: the sum is three and three squared is nine, while the squares are one and four, adding to five. The two results differ whenever both numbers are non-zero, and the size of the gap grows with the numbers. This is the misconception that Chapter 10 spends an entire lesson correcting, so it is worth killing now with a counterexample you can recall instantly.
Explain it
They are fluent with multiplication and have never seen a raised number.
Discussion prompt
In no more than four sentences, and without using the word exponent, explain what the small raised number in two to the third power is doing. Then give them one test they can run to check whether they have understood, and say what wrong answer the test is designed to catch.
Hint: The test should involve writing something out rather than working something out.
Answer:
A usable answer: the small raised number is a count. It tells you how many copies of the big number to multiply together — three copies here, so two times two times two. It is not one of the numbers being multiplied, and it never gets added to anything.
The test: write the multiplication out in full and count the copies. If they write two plus two plus two, or if they write two times three, the count is right but the operation is wrong — and both of those give six instead of eight, which is the single most common wrong answer at this stage.
Exit ticket
Name the weakest spot before you close the deck. That is the one worth ten minutes tonight.
Predict first
Which of these would you least want to be handed cold on a quiz tomorrow?
Correct: Whichever you picked is the right answer — and each one has a specific fix.
Why: Translating phrases is fixed by asking which word names the exponent and treating everything else as the base. Evaluating is fixed by writing the factors out and counting them before multiplying. Bracket questions are fixed by finishing the innermost grouping first and remembering that a bare exponent takes only the symbol in front of it. Units are fixed by matching the exponent on the unit to the exponent in the formula. Pick yours and do five of that kind tonight rather than twenty mixed ones.
Connect it up
Do this on paper. It is worth more than rereading the slides.
Draw it
At the top of a page write one power of your own choosing and label its three parts with arrows: base, exponent, and the whole power. Underneath, write that power out as repeated factors and draw a brace under the factors with the count written beside it. In the middle of the page, draw a square and a cube of the same edge, write the formula beside each and the unit its answer carries. At the bottom, write the pair 2x cubed and the quantity 2x all cubed side by side, evaluate both at x equal to 3, and box the one that is larger. Finally, in the margin, write the two-number counterexample that shows the square of a sum is not the sum of the squares.
The boxed expression at the bottom should be the bracketed one, and it should be eight times the other. If your two answers differ by any other factor, check whether you cubed the 2 in only one of them.
Recap
Five things, and the third one is the habit that prevents the two errors this lesson is really about.
| If the question says | Your first move is |
|---|---|
| Write in exponential form | Find the word that names the exponent |
| Evaluate the power | Substitute, then write out the factors |
| Evaluate 2x cubed | Notice there are no brackets, so only x is cubed |
| Evaluate the quantity 2x cubed | Finish the bracket first, then apply the exponent |
| Find the volume | Use the third power, and write cubic units |
Lesson 1.3 takes the grouping-symbol rule from this lesson and turns it into a complete order of operations, so that an expression with several different operations has exactly one correct value.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 1 Connections to Algebra — Lesson 1.2 Exponents and Powers §1.2, pp. 9-14 — everything on these slides traces back here
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