The three multiplication properties of exponents: the product of powers rule, which adds exponents; the power of a power rule, which multiplies them; and the power of a product rule, which distributes the exponent over the factors. Includes why each rule holds, telling them apart, and the difference between doubling a length and doubling an area.
Subject: Algebra 1 · 65 slides · symbolic lesson
Open the interactive version of this deck
Title
Algebra 1 · Chapter 8 — Exponents and Exponential Functions
Multiplication Properties of Exponents
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. 8 Exponents and Exponential Functions — Lesson 8.1 Multiplication Properties of Exponents §8.1, pp. 443-448 — the lesson these objectives are drawn from
Warm-up
Lesson 1.2 defined a power as repeated multiplication. This lesson uses that definition to derive three rules rather than to memorise them.
Discussion prompt
Write a squared times a cubed out as individual factors and count them. What single power have you got?
Hint: Two factors and then three more.
Answer:
\[ a^2 \cdot a^3 = (a \cdot a)(a \cdot a \cdot a) = a^5 \]
Five factors of a, so the answer is a to the fifth. The exponents added because the factors were counted, and every rule in this lesson comes out of that same counting.
Concept
An exponent records how many factors of the base are being multiplied. Every multiplication property of exponents follows from counting those factors, so the rules can be rebuilt rather than remembered.
product of powers property — To multiply powers with the same base, add the exponents: a to the m times a to the n is a to the m plus n.
The Developing Concepts investigation on page 441 arrives at the first rule by counting.
Figure (svg): Two powers written out as factors and counted together
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.1 Multiplication Properties of Exponents §8.1, pp. 441-443
Section
Section 1
Concept
To multiply powers that have the same base, add the exponents. The rule holds because the two exponents are counting factors of the same thing.
\[ a^m \cdot a^n = a^{m + n} \]
The bases must match; different bases have nothing in common to count.
Figure (svg): The product of powers property stated with an example
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.1 Multiplication Properties of Exponents §8.1, pp. 443-443 — the product of powers property and Example 1
Picture it
Add, do not multiply.
Figure (svg): The product of powers property stated with an example
Five cubed times five to the sixth is five to the ninth, not five to the eighteenth. Writing out a small case is the fastest way to check which operation the rule uses.
Worked example
This is Example 1 from the textbook.
\[ \text{Write as a single power: } \; 5^3 \cdot 5^6, \quad -2(-2)^4, \quad x^2 \cdot x^3 \cdot x^4. \]
Take the first
Why: The bases match, so add three and six.
\[ 5 ^{9} \]
Rewrite the second's first factor
Why: Negative two is negative two to the first power.
\[ (-2) ^{1}(-2) ^{4} \]
Add its exponents
Why: One plus four is five.
\[ (-2) ^{5} \]
Take the third
Why: Three exponents to add: two, three and four.
\[ x ^{9} \]
Figure (svg): A base with no visible exponent rewritten with an exponent of one
\[ 5^9, \quad (-2)^5, \quad x^9 \]
Verify: check the second by evaluating
Why: Negative two to the fourth is sixteen, and negative two times sixteen is negative thirty-two. Negative two to the fifth is also negative thirty-two, so the rule and the arithmetic agree — which is worth confirming once with a negative base.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.1 Multiplication Properties of Exponents §8.1, pp. 443-443
Sorting
The rule needs matching bases.
Sort into buckets
Sort each product by whether the product of powers rule applies.
Two of the six cannot be combined at all, and recognising that is as useful as knowing the rule — an expression with different bases is already simplified.
Worked example
Guided Practice 1 to 4. One has an invisible exponent.
\[ \text{Simplify } \; 4^2 \cdot 4^3, \quad (-3)^2(-3), \quad a \cdot a^7, \quad n^5 \cdot n^2 \cdot n^3. \]
Take the first
Why: Two plus three.
\[ 4 ^{5} \]
Take the second
Why: The second factor has an exponent of one.
\[ (-3) ^{3} \]
Take the third
Why: The a on its own is a to the first.
\[ a ^{8} \]
Take the fourth
Why: Five plus two plus three.
\[ n ^{10} \]
Figure (svg): The solution to Worked example four from guided practice shown as a ladder of expressions, one row per algebraic move
\[ 4^5, \quad (-3)^3, \quad a^8, \quad n^{10} \]
Verify: check the first by evaluating both sides
Why: Sixteen times sixty-four is one thousand and twenty-four, and four to the fifth is also one thousand and twenty-four. Checking a small case numerically is the way to confirm a rule you are unsure of, and it takes seconds.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.1 Multiplication Properties of Exponents §8.1, pp. 443-443
Trap
\[ 5^3 \cdot 5^6 \]
Multiply the exponents to get 5^18
Why: The expression contains a multiplication sign, so multiplying feels like the matching operation.
Writing it out gives three factors of five and then six more, which is nine factors. Five to the eighteenth is vastly larger than the truth.
\[ 5^3 \cdot 5^6 = 5^9 \]
Add the exponents, because the factors are being counted
Why: The multiplication is between the powers, and the counting is what the exponents record.
Testing on two squared times two cubed settles it: four times eight is thirty-two, which is two to the fifth rather than two to the sixth.
Faded example
The bases match.
Fill in the blanks
5^3 \cdot 5^6 = 5^6}} = 5^9}
Why: Three factors of five followed by six more gives nine factors in total, so the exponents add. Multiplying them would give five to the eighteenth, which is a completely different number.
Elimination
Simplify x squared times x cubed.
Eliminate the wrong options
Which is correct?
Survives elimination: A
Why: Two factors of x and then three more gives five factors. Testing with x equal to two settles it: four times eight is thirty-two, which is two to the fifth rather than two to the sixth.
Socratic
The rule is worth deriving rather than memorising.
Discussion prompt
Explain why multiplying powers with the same base adds their exponents, using what an exponent records. Then say why the rule fails when the bases differ.
Hint: Ask what the exponent is counting.
Answer:
An exponent counts how many factors of the base are being multiplied. Writing both powers out gives a run of m factors followed by a run of n factors, all of the same base, so the total is m plus n factors — which is what the exponent on the answer records.
With different bases the two runs are made of different things, so they cannot be merged into a single count. Two cubed times three to the fourth is eight times eighty-one, and there is no single base whose exponent describes that — the expression is already simplified.
Section
Section 2
Concept
To find a power of a power, multiply the exponents. Raising a squared to the third power means three copies of a squared, which is two added three times.
\[ (a^m)^n = a^{mn} \]
Adding a number n times is the same as multiplying by n, which is why this rule multiplies.
Figure (svg): The power of a power property shown by repeated multiplication
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.1 Multiplication Properties of Exponents §8.1, pp. 444-444 — the power of a power property and Example 2
Picture it
Adding two three times.
Figure (svg): The power of a power property shown by repeated multiplication
The rule is the product rule applied repeatedly, which is why one adds and the other multiplies. Neither has to be remembered separately once that connection is seen.
Worked example
This is Example 2 from the textbook.
\[ \text{Write as a single power: } \; (3^3)^2 \; \text{ and } \; (p^4)^4. \]
Take the first
Why: Multiply three by two.
\[ 3 ^{6} \]
Check by expanding
Why: Three cubed times three cubed is three to the sixth.
Take the second
Why: Multiply four by four.
\[ p ^{16} \]
Note the size
Why: Sixteen factors of p, from an expression with two fours in it.
Figure (svg): The power of a power property shown by repeated multiplication
\[ 3^6 \qquad p^{16} \]
Verify: evaluate the first both ways
Why: Three cubed is twenty-seven, and twenty-seven squared is seven hundred and twenty-nine. Three to the sixth is also seven hundred and twenty-nine, so the rule checks out on a case small enough to compute.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.1 Multiplication Properties of Exponents §8.1, pp. 444-444
Discrimination
Read what the outer exponent applies to.
Sort into buckets
Sort each expression by what happens to its exponents.
Worked example
Guided Practice 5 to 8. One has a negative base.
\[ \text{Simplify } \; (4^4)^3, \quad [(-3)^5]^2, \quad (n^4)^5, \quad (x^3)^3. \]
Take the first
Why: Four times three.
\[ 4 ^{12} \]
Take the second
Why: Five times two, with the base kept in brackets.
\[ (-3) ^{10} \]
Take the third
Why: Four times five.
\[ n ^{20} \]
Take the fourth
Why: Three times three.
\[ x ^{9} \]
Figure (svg): The solution to Worked example four from guided practice shown as a ladder of expressions, one row per algebraic move
\[ 4^{12}, \quad (-3)^{10}, \quad n^{20}, \quad x^9 \]
Verify: think about the sign of the second
Why: Ten is even, so negative three to the tenth is positive. Squaring anything gives a positive result, and the outer exponent of two is what guarantees it — the brackets around the negative base are what make that visible.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.1 Multiplication Properties of Exponents §8.1, pp. 444-444
Error analysis
The student simplified two exponent expressions.
Annotate
On: \( \begin{aligned} a^2 \cdot a^3 &= a^6 \\ (a^2)^3 &= a^5 \end{aligned} \)
The two expressions look alike and behave differently. Reading whether the outer exponent applies to a whole power or sits alongside a separate factor is what distinguishes them.
Faded example
A power of a power.
Fill in the blanks
(3^3)^2 = 3^2}} = 3^6}
Why: Two copies of three cubed gives three to the sixth, because adding three twice is multiplying by two. Evaluating both sides gives seven hundred and twenty-nine, confirming the rule on a small case.
Prediction
The base is negative and the outer exponent is even.
Predict first
Is [(-3)^5]^2 positive or negative?
Correct: Positive, since the total exponent 10 is even.
\[ [(-3)^5]^2 = (-3)^{10} > 0 \]
Why: An even number of negative factors gives a positive product, and the total exponent is five times two, which is ten. The inner exponent being odd matters only within the brackets; squaring whatever it produced makes the result positive regardless. Any expression squared is non-negative, which settles it without any arithmetic.
Socratic
The other rule added.
Discussion prompt
Explain why raising a power to a power multiplies the exponents, deriving it from the product rule. Then say what would happen with three levels of exponent.
Hint: Write out the outer power as repeated multiplication.
Answer:
Raising a squared to the third power means a squared times a squared times a squared. The product rule adds those exponents, giving two plus two plus two — and adding two three times is the same as multiplying two by three. So the second rule is the first applied repeatedly.
Three levels would multiply all three exponents. Taking a squared, cubing it and then raising that to the fourth gives twenty-four factors, since two times three times four is twenty-four. Each new level multiplies again, which is why towers of exponents grow so quickly.
Section
Section 3
Concept
To find a power of a product, find the power of each factor and multiply. The exponent distributes over the factors because multiplication can be regrouped freely.
\[ (ab)^n = a^n b^n \]
Every factor inside the brackets receives the exponent, including numbers.
Figure (svg): The power of a product property expanded and regrouped
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.1 Multiplication Properties of Exponents §8.1, pp. 444-444 — the power of a product property and Example 3
Picture it
Collect the a's, then the b's.
Figure (svg): The power of a product property expanded and regrouped
The regrouping is allowed by the commutative and associative properties from Chapter 2, which is why the rule needs no separate justification.
Worked example
This is Example 3 from the textbook.
\[ \text{Simplify } \; (-6 \cdot 5)^2 \; \text{ and } \; (4yz)^3. \]
Distribute the exponent in the first
Why: Each factor is squared.
\[ (-6) ^{2} 5 ^{2} \]
Evaluate
Why: Thirty-six times twenty-five.
\[ 900 \]
Distribute in the second
Why: The four, the y and the z each get the exponent.
\[ 4 ^{3} y ^{3} z ^{3} \]
Evaluate the numerical part
Why: Four cubed is sixty-four.
\[ 64 y ^{3} z ^{3} \]
Figure (svg): The power of a product property expanded and regrouped
\[ 900 \qquad 64y^3z^3 \]
Verify: check the first the other way
Why: Negative six times five is negative thirty, and negative thirty squared is nine hundred — the same answer. Multiplying first and then squaring, or squaring first and then multiplying, must agree, which is exactly what the rule asserts.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.1 Multiplication Properties of Exponents §8.1, pp. 444-444
Faded example
Every factor inside.
Fill in the blanks
(4yz)^3 = 4^3 y^3 z^3 = 64y^3z^3
Why: The four receives the exponent along with the y and the z, giving sixty-four rather than four. Leaving the coefficient unraised is the standard error here and it changes the answer by a factor of sixteen.
Worked example
Guided Practice 9 to 12. Two numerical and two with variables.
\[ \text{Simplify } \; (2 \cdot 4)^3, \quad (-3 \cdot 5)^2, \quad (2w)^6, \quad (-7a)^2. \]
Take the first
Why: Eight cubed, or two cubed times four cubed.
\[ 512 \]
Take the second
Why: Negative fifteen squared.
\[ 225 \]
Take the third
Why: Two to the sixth times w to the sixth.
\[ 64 w ^{6} \]
Take the fourth
Why: Negative seven squared times a squared.
\[ 49 a ^{2} \]
Figure (svg): The solution to Worked example four from guided practice shown as a ladder of expressions, one row per algebraic move
\[ 512, \quad 225, \quad 64w^6, \quad 49a^2 \]
Verify: check the coefficient in the third
Why: Two to the sixth is sixty-four, not twelve. The coefficient is raised to the power rather than multiplied by it, which is the commonest error with this rule and is worth confirming numerically.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.1 Multiplication Properties of Exponents §8.1, pp. 444-444
Trap
\[ (2w)^6 \]
Write 2w^6, raising only the variable
Why: The exponent looks like it belongs to the letter next to it.
The brackets mean the whole product is raised, so the two is raised as well. Two to the sixth is sixty-four, so the answer is sixty-four w to the sixth — thirty-two times larger than the version written.
\[ (2w)^6 = 2^6 w^6 = 64w^6 \]
Give the exponent to every factor inside the brackets
Why: That is what the power of a product rule says.
Testing with w equal to one settles it: the original is two to the sixth, which is sixty-four, and the wrong version gives two.
Elimination
Simplify the quantity 3x, raised to the fourth power.
Eliminate the wrong options
Which is correct?
Survives elimination: A
Why: Three to the fourth is eighty-one and x to the fourth comes from the same exponent, so the answer is eighty-one x to the fourth. Testing with x equal to one gives eighty-one, which rejects the other three immediately.
Sorting
Read the structure of each expression.
Sort into buckets
Sort each expression by the property that simplifies it.
Identifying the structure decides which rule applies, and the three structures look different once you know what to look for: two powers, one power in brackets, or several factors in brackets.
Socratic
The rule rearranges a long product.
Discussion prompt
Explain which properties of multiplication allow the a's and b's to be collected separately when a product is raised to a power. Then say why the rule would fail for a sum inside the brackets.
Hint: Think about Chapter 2's properties.
Answer:
Multiplication is commutative and associative, so the factors of a long product can be reordered and regrouped freely without changing the value. That is what lets every a be gathered together and every b likewise, producing a to the n times b to the n.
A sum cannot be regrouped that way. The quantity a plus b, squared, is not a squared plus b squared — expanding it gives a squared plus 2ab plus b squared, and the cross terms have nowhere to go. Chapter 10 develops that expansion, and the difference between it and this rule is worth noticing now.
Section
Section 4
Concept
The three rules apply to three different structures. Two separate powers multiplied: add. One power raised again: multiply. A product raised to a power: distribute.
Brackets are the signal that distinguishes the last two from the first.
Figure (svg): The three multiplication properties of exponents side by side
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.1 Multiplication Properties of Exponents §8.1, pp. 443-448 — the three properties stated across the lesson
Picture it
Add, multiply, distribute.
Figure (svg): The three multiplication properties of exponents side by side
Every problem in the lesson is one of these three, or a combination of them. Naming the structure before touching the exponents is what keeps them apart.
Worked example
Naming the structure decides the action.
\[ \text{Simplify } \; x^2 \cdot x^5, \quad (x^2)^5, \quad (x^2 y)^5. \]
Take the first
Why: Two separate powers, so add.
\[ x ^{7} \]
Take the second
Why: One power raised again, so multiply.
\[ x ^{10} \]
Take the third
Why: A product raised to a power, so distribute.
\[ (x ^{2}) ^{5} y ^{5} \]
Finish the third
Why: Then multiply the inner exponents.
\[ x ^{10} y ^{5} \]
Figure (svg): Two columns contrasting the product rule with the power rule
\[ x^7, \quad x^{10}, \quad x^{10}y^5 \]
Verify: notice the third used two rules
Why: Distributing gave x squared raised to the fifth, and then the power of a power rule finished it. Combining rules is normal, and naming each structure as it appears is what keeps the sequence straight.
Matching
Read what the exponent applies to.
Match the pairs
Why: The four expressions differ only in their brackets and coefficients, and each needs a different combination of rules. The last needs two: distribute the exponent, then multiply the inner exponents.
Worked example
Several rules in one problem.
\[ \text{Simplify } \; (2a^3)^2 \cdot a^4. \]
Distribute the outer exponent
Why: The two and the a cubed each get squared.
\[ 2 ^{2}(a ^{3}) ^{2} a ^{4} \]
Apply the power of a power rule
Why: Three times two is six.
\[ 4 a ^{6} a ^{4} \]
Apply the product rule
Why: Six plus four is ten.
\[ 4 a ^{10} \]
State the answer
Why: Four a to the tenth.
\[ 4 a ^{10} \]
Figure (svg): The solution to Worked example a combined expression shown as a ladder of expressions, one row per algebraic move
\[ 4a^{10} \]
Verify: check with a equal to one
Why: The original gives two squared times one, which is four, and the answer gives four. Substituting one strips out the variable and tests the coefficient alone, which is where the power of a product rule is most often mishandled.
Trap
\[ (2a^3)^2 \]
Write 2a^6, squaring the a cubed and leaving the two
Why: The exponent looks like it belongs with the power next to it.
Both factors inside the brackets receive the exponent, so the two becomes four. Testing with a equal to one gives four in the original and two in the written version.
\[ (2a^3)^2 = 2^2 (a^3)^2 = 4a^6 \]
Distribute first, then simplify each factor
Why: The power of a product rule comes before the power of a power rule here.
Writing the distribution as a separate line, with every factor shown, makes an omitted factor visible.
Faded example
Distribute, then multiply.
Fill in the blanks
(2a^3)^2 = 2^2 (a^3)^2 = 4a^6}
Why: The power of a product rule gives the exponent to both factors, and then the power of a power rule handles the inner exponent. Doing them as separate lines is what stops one of the two factors being missed.
Elimination
Three of these do not.
Eliminate the wrong options
Which one is x to the sixth?
Survives elimination: A
Why: Two times three is six, so the first is x to the sixth. Option C is worth naming because sums of powers cannot be simplified at all — every rule in this lesson is about products.
Socratic
Every property here concerns multiplication.
Discussion prompt
Explain why x squared plus x cubed cannot be written as a single power, and why the exponent rules say nothing about it. Then say what x squared plus x squared does simplify to, and why that is different.
Hint: Ask what the rules are counting.
Answer:
The rules count factors, and a sum is not a product — nothing is being multiplied, so there is nothing to count together. Two factors of x and three factors of x are two separate quantities being added, and adding them does not produce a run of factors of any length.
Two identical terms do combine: x squared plus x squared is two x squared, which is combining like terms from Lesson 2.7 rather than an exponent rule. The exponent stays at two and a coefficient appears, which is the opposite of what would happen if they were multiplied. Distinguishing those two situations is worth the second it takes.
Section
Section 5
Concept
Because area formulas contain a squared length, doubling a radius does not double an area. The power of a product rule turns two r into four r squared.
\[ \pi(2r)^2 = \pi \cdot 4r^2 = 4\pi r^2 \]
Twice as big in radius means four times as big in area.
Figure (svg): Two circles whose radii differ by a factor of two, with areas differing by four
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.1 Multiplication Properties of Exponents §8.1, pp. 445-445 — Example 5 on the areas of two irrigation circles
Picture it
Double the radius, quadruple the area.
Figure (svg): Two circles whose radii differ by a factor of two, with areas differing by four
The chapter opener asks what twice as big means for a circle. The answer depends entirely on whether the radius or the area is being doubled, and the two are not the same claim.
Worked example
This is the situation in Example 5 of the textbook.
\[ \text{One irrigation circle has radius } r \text{ and another } 2r. \text{ Compare their areas.} \]
Write the first area
Why: Pi times r squared.
\[ \pi r ^{2} \]
Write the second
Why: Pi times the quantity 2r, squared.
\[ \pi(2 r) ^{2} \]
Apply the power of a product rule
Why: The two and the r are both squared.
\[ \pi(4 r ^{2}) \]
Compare
Why: Four times the first area.
\[ 4 \times \]
Figure (svg): Two circles whose radii differ by a factor of two, with areas differing by four
\[ \pi(2r)^2 = 4\pi r^2 \]
Verify: check with a number
Why: A circle of radius three has area nine pi and one of radius six has area thirty-six pi, which is four times as much. The factor is four rather than two because the radius appears squared in the formula.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.1 Multiplication Properties of Exponents §8.1, pp. 445-445
Prediction
The radius of a circle is tripled.
Predict first
By what factor does the area change?
Correct: 9, since the radius is squared in the formula.
\[ \pi(3r)^2 = 9\pi r^2 \]
Why: Pi times the quantity 3r, squared, is nine pi r squared by the power of a product rule. The factor is the cube of nothing and the square of three, because area depends on the square of a length. And it does not depend on the original radius: the factor of nine appears whatever r was.
Worked example
The chapter opener's question, answered.
\[ \text{To double the area of a circle, by what factor must the radius grow?} \]
Write the requirement
Why: The new area is twice the old.
\[ \pi(k r) ^{2} = 2 \pi r ^{2} \]
Apply the rule
Why: k squared r squared on the left.
\[ k ^{2} = 2 \]
Solve for the factor
Why: k is the square root of two, about 1.41.
\[ \text{about } 1.41 \]
Interpret
Why: The radius grows by about forty-one per cent, not by a hundred.
\[ 1.41 \times \]
Figure (svg): The solution to Worked example what twice as big means shown as a ladder of expressions, one row per algebraic move
\[ k = \sqrt{2} \approx 1.41 \]
Verify: check the factor
Why: A radius of 1.41 gives an area of about 1.99 pi against one pi, which is very nearly double. Doubling the radius would have quadrupled the area instead, so the two versions of twice as big differ substantially.
Trap
One circle is twice as big as another.
Take that to mean the radius doubles and the area doubles too
Why: Twice as big sounds like a single unambiguous statement.
Doubling the radius quadruples the area, so the two readings describe different circles. The phrase has to be pinned to a quantity before it means anything.
Ask which quantity is being doubled before answering
Why: Radius, area and volume all scale differently under the same enlargement.
The power in the formula decides the factor: a length scales by k, an area by k squared and a volume by k cubed.
Faded example
The exponent goes to both factors.
Fill in the blanks
\pi(2r)^2 = \pi \cdot 4 \cdot r^2 = 4\pi r^2
Why: Squaring the whole quantity 2r squares both the two and the r, giving four r squared. The area therefore grows by a factor of four, not two — which is the power of a product rule doing something with a physical consequence.
Sorting
The lengths are all doubled.
Sort into buckets
Sort each quantity by the factor it grows by.
The exponent in the formula is exactly the exponent on the scaling factor, which is the power of a product rule appearing as a fact about shapes.
Socratic
One rule explains all three columns.
Discussion prompt
Explain why doubling a length multiplies an area by four and a volume by eight, using the power of a product rule. Then say what would happen to a quantity depending on the fourth power of a length.
Hint: Look at where the length sits in each formula.
Answer:
An area formula contains a length squared, so replacing that length by 2L gives the quantity 2L squared, which the rule expands as four L squared. A volume formula contains a cube, so 2L cubed gives eight L cubed. The exponent in the formula becomes the exponent on the two.
A quantity depending on the fourth power would grow by two to the fourth, which is sixteen. That is not a curiosity — the resistance of a beam and the power output of some systems depend on high powers of a dimension, so a modest enlargement produces an enormous change, and the power of a product rule is what predicts it.
Comparison
Fill the blanks from memory before you scroll back.
Comparison matrix
| Property | Looks like | Do this to the exponents |
|---|---|---|
| Product of powers | a^m x a^n | add them |
| Power of a power | (a^m)^n | multiply them |
| Power of a product | (ab)^n | give the exponent to each factor |
The middle column is what tells you which row you are in, and the brackets are the feature that separates the last two rows from the first.
Pattern
Whether the expression uses one rule or three, the same five moves cover it.
Step five's substitution strips out the variables and tests the numerical part alone, which is where the power of a product rule is most often mishandled.
OpenStax Elementary Algebra 2e, §6.2 Use Multiplication Properties of Exponents §6.2
Check
Two separate powers.
Check your understanding
Simplify x^4 times x^6.
Answer: A
Why: Four factors of x followed by six more gives ten factors, so the exponents add. Testing with x equal to two gives sixteen times sixty-four, which is one thousand and twenty-four, matching two to the tenth.
Check
A power raised again.
Check your understanding
Simplify (n^3) raised to the fourth power.
Answer: A
Why: Four copies of n cubed gives twelve factors, so the exponents multiply. Writing it as n cubed times n cubed times n cubed times n cubed and adding confirms it.
Check
Every factor inside the brackets.
Check your understanding
Simplify the quantity 5x, raised to the third power.
Answer: A
Why: Both factors receive the exponent, so five cubed is one hundred and twenty-five and x cubed comes from the same exponent. Substituting x equal to one gives one hundred and twenty-five, which confirms the coefficient.
Real world
This is the chapter opener's question. Two circular irrigation fields are watered by rotating sprinklers, and a farmer describes one as twice as big as the other.
Discussion prompt
Say what twice as big could mean, work out the relationship between the two radii in each case, and say which interpretation a farmer buying seed would care about.
Hint: Area is proportional to the square of the radius.
Answer:
\[ \text{twice the radius}: \; \pi(2r)^2 = 4\pi r^2 \quad \text{four times the area} \]
\[ \text{twice the area}: \; \pi(kr)^2 = 2\pi r^2 \;\Longrightarrow\; k = \sqrt{2} \approx 1.41 \]
Doubling the radius quadruples the area, while doubling the area needs the radius to grow by only about forty-one per cent. The two readings of twice as big describe noticeably different fields.
A farmer buying seed cares about the area, since seed is spread over ground rather than along a radius. So twice as big should mean twice the area — and a sprinkler arm only forty-one per cent longer achieves it, which is a much smaller change to the equipment than doubling the arm would be.
Commit first
Answer, then rate your confidence honestly. Confident and wrong is the combination worth finding.
Predict first
What is (2a^3) squared?
Correct: 4a^6, since both factors are squared.
\[ (2a^3)^2 = 2^2 (a^3)^2 = 4a^6 \]
\[ \text{at } a = 1: \; (2)^2 = 4 \;\checkmark \]
Why: The brackets mean the whole product is raised, so the two becomes four and the a cubed becomes a to the sixth. The first option leaves the coefficient unraised, which is the commonest error with this rule and is caught immediately by substituting a equal to one: the original gives four and that version gives two. The exponents multiply rather than adding, since a power is being raised to a power.
Explain it
They know what a power is and keep confusing the three rules.
Discussion prompt
In no more than four sentences, explain how to tell the three rules apart without memorising them. Then give them the check that catches a wrong one.
Hint: Write the factors out.
Answer:
A usable answer: an exponent just counts how many times the base is multiplied, so write the factors out and count. Two powers side by side lay the runs end to end, so the counts add; a power in brackets with another exponent takes several copies of the whole run, so the counts multiply; and a product in brackets gives the exponent to every factor inside.
To check, put a small number in — two usually works — and evaluate both your answer and the original. If they do not match you used the wrong rule, and doing that once on a small case is quicker than trying to remember which rule was which.
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: Adding against multiplying is fixed by writing out a small case and counting factors. The coefficient is fixed by substituting one for every variable and checking the number that remains. Invisible exponents are fixed by writing them in before starting. Combining rules is fixed by doing brackets first and writing each rule on its own line. 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 out a squared times a cubed as individual factors, count them, and write the rule it demonstrates. Underneath, do the same for a squared raised to the third power and for the quantity ab cubed, in each case expanding fully before writing the rule. In the middle, simplify four expressions using the three rules — one of each and one combining two of them — writing beside each line which rule you used. Substitute one for every variable in your combined expression and check the numerical part against the original. In the lower half, draw two circles, one with twice the radius of the other, and compute both areas symbolically to show the factor of four, then work out by what factor the radius would have to grow to double the area instead. Finally, in the margin, write the three rules as a table with a column for what happens to the exponents.
Your substitution check should give the same number for the original and the simplified version. If it does not, the coefficient was mishandled — that is what setting every variable to one isolates.
Recap
Five things, and the second is where the coefficient goes missing.
| If the question says | Your first move is |
|---|---|
| Two powers with the same base | Add the exponents |
| A power inside brackets with an exponent | Multiply the exponents |
| A product inside brackets | Give the exponent to every factor |
| A base with no exponent shown | Write the exponent as 1 |
| The radius doubles | The area is multiplied by 4 |
Lesson 8.2 extends the exponents beyond the counting numbers. An exponent of zero and negative exponents both have to be defined so that the product rule keeps working, which turns out to force exactly one choice for each.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.1 Multiplication Properties of Exponents §8.1, pp. 443-448 — everything on these slides traces back here
Want this taught 1-on-1? Alexander tutors Algebra 1 — $55/session, free consultation.