Writing numbers as a coefficient between one and ten times a power of ten. Includes converting in both directions by moving the decimal point, reading the exponent's sign as large or small, and multiplying, dividing and raising such numbers to powers using the exponent rules with a final adjustment back into standard form.
Subject: Algebra 1 · 65 slides · symbolic lesson
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Title
Algebra 1 · Chapter 8 — Exponents and Exponential Functions
Scientific Notation
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 8.5 Scientific Notation §8.5, pp. 469-474 — the lesson these objectives are drawn from
Warm-up
Lesson 8.4 divided powers by subtracting exponents. This lesson uses that to make arithmetic with very large and very small numbers manageable.
Discussion prompt
Compute eighty million divided by twenty thousand, first by long division and then by writing both as a coefficient times a power of ten.
Hint: Divide the coefficients and subtract the exponents.
Answer:
\[ \dfrac{8 \times 10^7}{2 \times 10^4} = 4 \times 10^3 = 4000 \]
The long division takes some care and the second route is two small steps. Writing numbers as a coefficient times a power of ten turns hard arithmetic into easy arithmetic, which is the whole reason the notation exists.
Concept
A number is written in scientific notation when it has the form c times ten to the n, where c is at least one and less than ten and n is an integer. Every number has exactly one such representation.
scientific notation — The form c times ten to the n, where the coefficient c is at least one and less than ten and the exponent n is an integer.
The restriction on the coefficient is what makes the representation unique.
Figure (svg): The form of a number in scientific notation with its two parts labelled
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.5 Scientific Notation §8.5, pp. 469-469
Section
Section 1
Concept
Scientific notation writes a number as a coefficient times a power of ten, with the coefficient at least one and less than ten — so exactly one nonzero digit sits before the decimal point.
\[ c \times 10^n, \quad 1 \le c < 10 \]
The exponent may be positive, zero or negative.
Figure (svg): The form of a number in scientific notation with its two parts labelled
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.5 Scientific Notation §8.5, pp. 469-469 — the definition of scientific notation
Picture it
A coefficient and an exponent.
Figure (svg): The form of a number in scientific notation with its two parts labelled
The restriction on the coefficient is not decoration. Without it a number could be written in many ways, and comparing two numbers would need converting first.
Worked example
The coefficient must lie between one and ten.
\[ \text{Which are in scientific notation? } \; 3.4 \times 10^4, \; 34 \times 10^3, \; 0.34 \times 10^5, \; 5.6 \times 10^0. \]
Check the first
Why: Three point four is between one and ten.
Check the second
Why: Thirty-four is at least ten.
Check the third
Why: Zero point three four is below one.
Check the fourth
Why: Five point six is in range, and a zero exponent is allowed.
Figure (svg): The solution to Worked example which are in scientific notation shown as a ladder of expressions, one row per algebraic move
\[ 3.4 \times 10^4 \text{ and } 5.6 \times 10^0 \]
Verify: check that all four are the same kinds of number
Why: The first three all equal thirty-four thousand, written three different ways. Only one of the three is in scientific notation, which is exactly what the restriction on the coefficient achieves — one number, one form.
Sorting
Check the coefficient.
Sort into buckets
Sort each expression by whether it is written in scientific notation.
The last item is the boundary case worth noticing. Ten is not less than ten, so a coefficient of exactly ten is not allowed and must be written as one times ten to the next power.
Worked example
Adjusting the coefficient adjusts the exponent.
\[ \text{Rewrite } \; 34 \times 10^3 \; \text{ and } \; 0.34 \times 10^5 \; \text{ correctly.} \]
Take the first
Why: The coefficient is too large, so move the point left.
\[ 3.4 \]
Compensate
Why: One move left raises the exponent by one.
\[ 3.4 x 10 ^{4} \]
Take the second
Why: The coefficient is too small, so move the point right.
\[ 3.4 \]
Compensate
Why: One move right lowers the exponent by one.
\[ 3.4 x 10 ^{4} \]
Figure (svg): A result adjusted back into scientific notation in both directions
\[ 3.4 \times 10^4 \]
Verify: check that the value did not change
Why: All three expressions equal thirty-four thousand. Moving the point and changing the exponent are compensating changes, so the value stays fixed while the written form is corrected.
Trap
\[ 34 \times 10^3 \]
Report this as scientific notation, since it has a power of ten in it
Why: The shape looks right and the value is correct.
The coefficient must be at least one and less than ten, and thirty-four is not. The value is right and the form is not, which matters because the form is what makes comparisons immediate.
\[ 3.4 \times 10^4 \]
Move the point until exactly one nonzero digit is in front of it, adjusting the exponent to compensate
Why: The two changes cancel, so the value is preserved.
Checking that the coefficient is between one and ten is the last step of every conversion.
Faded example
At least one, less than ten.
Fill in the blanks
In 3.4 times ten to the fourth the coefficient is 3.4, which is between 1 and 10, so the form is correct.
Why: The coefficient sits in the allowed range, so exactly one nonzero digit is before the decimal point. Any number outside that range has to be adjusted, with a compensating change to the exponent.
Elimination
All four have the right value.
Eliminate the wrong options
Which is correctly written?
Survives elimination: A
Why: All four equal thirty-four thousand and only one has a coefficient between one and ten. That is what makes the representation unique, which is the property the notation is designed to have.
Socratic
The value would be right either way.
Discussion prompt
Explain what is gained by insisting that the coefficient lies between one and ten. Then say what would be lost if any coefficient were allowed.
Hint: Ask how you would compare two numbers.
Answer:
The restriction makes the representation unique, so every number has exactly one scientific form. That means two numbers can be compared by looking at their exponents first and their coefficients second, without any conversion — which is the notation's main practical benefit.
Without it, thirty-four thousand could be written as 34 times ten cubed, 0.34 times ten to the fifth or 340 times ten squared, and comparing two such expressions would require rewriting them into a common form first. The restriction does that rewriting once and for all.
Section
Section 2
Concept
To write a number in decimal form, move the decimal point by as many places as the exponent says — to the right when the exponent is positive and to the left when it is negative.
The exponent is exactly the number of places.
Figure (svg): A decimal point moved right for a positive exponent and left for a negative one
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.5 Scientific Notation §8.5, pp. 469-469 — Example 1 and its Study Tip on which way to move the point
Picture it
The sign gives the direction.
Figure (svg): A decimal point moved right for a positive exponent and left for a negative one
Multiplying by ten moves the point one place right and dividing moves it one left, so the exponent is a running total of those single moves.
Worked example
This is Example 1 from the textbook.
\[ \text{Write in decimal form: } \; 2.83 \times 10^1, \; 4.9 \times 10^5, \; 8 \times 10^{-1}, \; 1.23 \times 10^{-3}. \]
Take the first
Why: Move one place right.
\[ 28.3 \]
Take the second
Why: Move five places right, filling with zeros.
\[ 490 000 \]
Take the third
Why: Move one place left.
\[ 0.8 \]
Take the fourth
Why: Move three places left.
\[ 0.00123 \]
Figure (svg): A decimal point moved right for a positive exponent and left for a negative one
\[ 28.3, \quad 490\,000, \quad 0.8, \quad 0.00123 \]
Verify: check each against the exponent's sign
Why: The two positive exponents produced numbers of at least ten and the two negative ones produced numbers below one. That correspondence holds always, so it is a fast check on which way the point was moved.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.5 Scientific Notation §8.5, pp. 469-469
Translation
Move the point by the exponent.
Match the pairs
Why: Positive exponents give numbers of at least ten and negative ones give numbers below one, with the exponent counting the places moved. Checking the size before converting catches a swapped direction immediately.
Worked example
Guided Practice 1 to 6. Three large and three small.
\[ \text{Convert } \; 2.39 \times 10^4, \; 1.045 \times 10^7, \; 3.7 \times 10^{-8}, \; 8.4 \times 10^{-6}, \; 1.0 \times 10^2, \; 9.2 \times 10^8. \]
Take the positive exponents
Why: Four, seven, two and eight places right.
Write them out
Why: 23 900, 10 450 000, 100 and 920 000 000.
Take the negative exponents
Why: Eight and six places left.
Write them out
Why: 0.000000037 and 0.0000084.
Figure (svg): The solution to Worked example six from guided practice shown as a ladder of expressions, one row per algebraic move
\[ 23\,900; \; 10\,450\,000; \; 0.000000037 \]
Verify: count the zeros in one of the small ones
Why: For 3.7 times ten to the negative eight, the point moves eight places left, so there are seven zeros between the point and the three. Counting the places rather than the zeros avoids being one out, since the digit itself occupies the eighth place.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.5 Scientific Notation §8.5, pp. 469-469
Error analysis
The student converted two numbers to decimal form.
Annotate
On: \( \begin{aligned} 4.9 \times 10^5 &= 0.000049 \\ 1.23 \times 10^{-3} &= 1230 \end{aligned} \)
Before converting, ask whether the answer should be large or small. The exponent's sign settles that in a glance, and it makes a swapped direction obvious.
Faded example
The exponent is the count.
Fill in the blanks
In 1.23 times ten to the negative three, the point moves 3 places to the left, giving 0.00123.
Why: Three places to the left puts two zeros between the point and the one. The exponent gives the count and its sign gives the direction, which is the whole conversion.
Prediction
Read the exponent's sign first.
Predict first
Is 9.2 times ten to the negative eight large or small?
Correct: Small: less than one.
\[ 9.2 \times 10^{-8} = 0.000000092 \]
Why: A negative exponent moves the point left, so the value is 0.000000092 — far below one. The coefficient is always between one and ten, so it can never by itself make a number large or small; only the exponent does that, which is why its sign is the first thing to read.
Socratic
The connection is worth stating.
Discussion prompt
Explain why multiplying by ten to the n moves the decimal point n places. Then say why the number system makes this true and what would change in a base other than ten.
Hint: Ask what multiplying by ten does to a digit's position.
Answer:
In our number system each place is worth ten times the one to its right, so multiplying by ten promotes every digit one place — which is the same thing as moving the point one place right. Doing that n times moves it n places, and dividing does the reverse.
It is true because the notation is base ten, so ten is the number that shifts places. In base two, multiplying by two would shift the binary point, and powers of two would play this role — which is why computers use a binary version of the same idea for very large and very small numbers.
Section
Section 3
Concept
To convert a decimal number, move the point until exactly one nonzero digit is in front of it, then count the moves. Moving left gives a positive exponent and moving right a negative one.
A number already between one and ten gets an exponent of zero.
Figure (svg): A decimal number converted into scientific notation
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.5 Scientific Notation §8.5, pp. 470-470 — Example 2, Write Numbers in Scientific Notation
Picture it
Four places left, so a fourth power.
Figure (svg): A decimal number converted into scientific notation
Making the coefficient smaller must be compensated by making the power larger, which is why moving left gives a positive exponent. The two changes cancel exactly.
Worked example
This is Example 2 from the textbook.
\[ \text{Write in scientific notation: } \; 34\,000, \; 1.78, \; 0.0007. \]
Take the first
Why: Move the point four places left to get 3.4.
\[ 3.4 x 10 ^{4} \]
Take the second
Why: It is already between one and ten.
\[ 1.78 x 10 ^{0} \]
Take the third
Why: Move four places right to get 7.
\[ 7 x 10 ^{-4} \]
Check the signs
Why: Left gave positive and right gave negative.
Figure (svg): A decimal number converted into scientific notation
\[ 3.4 \times 10^4, \quad 1.78 \times 10^0, \quad 7 \times 10^{-4} \]
Verify: convert one back
Why: Moving the point four places right in 3.4 gives thirty-four thousand, which is where we started. Converting back is the natural check and it uses the previous section's method.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.5 Scientific Notation §8.5, pp. 470-470
Faded example
Below one means a negative exponent.
Fill in the blanks
0.0007 = 7 \times 10^-4} \qquad 34\,000 = 3.4 \times 10^4}
Why: The two conversions move the same number of places in opposite directions, giving exponents of the same size and opposite signs. Matching the sign to whether the number is large or small is the fastest check.
Worked example
Guided Practice 7 to 10. Two large and two small.
\[ \text{Convert } \; 423, \; 2\,000\,000, \; 0.0001, \; 0.0098. \]
Take 423
Why: Two places left.
\[ 4.23 x 10 ^{2} \]
Take two million
Why: Six places left.
\[ 2 x 10 ^{6} \]
Take 0.0001
Why: Four places right.
\[ 1 x 10 ^{-4} \]
Take 0.0098
Why: Three places right.
\[ 9.8 x 10 ^{-3} \]
Figure (svg): The solution to Worked example four from guided practice shown as a ladder of expressions, one row per algebraic move
\[ 4.23 \times 10^2, \; 2 \times 10^6, \; 10^{-4}, \; 9.8 \times 10^{-3} \]
Verify: check the third's exponent by counting
Why: In 0.0001 the first nonzero digit is in the fourth decimal place, so the point moves four places right and the exponent is negative four. Counting to the first nonzero digit rather than counting zeros avoids the off-by-one error.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.5 Scientific Notation §8.5, pp. 470-470
Trap
\[ 0.0007 \]
Move the point four places right and write 7 times ten to the fourth
Why: The count is right and the direction of the move is copied into the sign.
Seven times ten to the fourth is seventy thousand, not seven ten-thousandths. Making the coefficient larger has to be compensated by making the power smaller, so the exponent is negative four.
\[ 0.0007 = 7 \times 10^{-4} \]
Ask whether the original number is large or small, and match the sign to that
Why: Below one means a negative exponent.
The compensation idea is the reliable version: whichever way the coefficient changes, the exponent changes the other way.
Sorting
Compare the number with one.
Sort into buckets
Sort each number by the sign of its exponent in scientific notation.
The dividing line is between one and ten, where the exponent is zero. Everything above gets a positive exponent and everything below a negative one.
Elimination
Convert 0.0098 to scientific notation.
Eliminate the wrong options
Which is correct?
Survives elimination: A
Why: The point moves three places right to reach 9.8, so the exponent is negative three. Options C and D have the right value and the wrong form, which is why checking the coefficient's range is the last step of every conversion.
Socratic
The value must not change.
Discussion prompt
Explain why moving the decimal point left requires the exponent to increase. Then say what that means about the total effect of the two changes.
Hint: Ask what each change does to the value.
Answer:
Moving the point left divides the coefficient by ten for each place moved, which makes the number smaller. To leave the value unchanged, something must multiply it back by ten for each move — and raising the exponent by one does exactly that.
So the two changes multiply and divide by the same amount, and their net effect on the value is nothing at all. Only the written form changes, which is why conversion is always reversible and why checking by converting back always works.
Section
Section 4
Concept
To multiply or divide numbers in scientific notation, combine the coefficients and combine the powers of ten using the exponent rules, then adjust the result back into scientific notation.
The regrouping is allowed by the commutative and associative properties.
Figure (svg): Two numbers in scientific notation multiplied by regrouping
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.5 Scientific Notation §8.5, pp. 470-470 — Example 3, Operations with Scientific Notation
Picture it
Regroup, combine, adjust.
Figure (svg): Two numbers in scientific notation multiplied by regrouping
The final adjustment is not optional. Ten point six four is outside the allowed range, so one more move is needed before the answer is in scientific notation.
Worked example
This is Example 3, parts a and b, from the textbook.
\[ \text{Compute } \; (1.4 \times 10^4)(7.6 \times 10^3) \; \text{ and } \; \dfrac{1.2 \times 10^1}{4.8 \times 10^4}. \]
Regroup the product
Why: Coefficients together, powers together.
\[ (1.4) (7.6) x(10 ^{4}) (10 ^{3}) \]
Combine
Why: Ten point six four times ten to the seventh.
\[ 10.64 x 10 ^{7} \]
Adjust
Why: The coefficient is too large, so move left and raise the exponent.
\[ 1.064 x 10 ^{8} \]
Do the quotient
Why: Coefficients divide to 0.25 and exponents subtract to negative three.
\[ 2.5 x 10 ^{-4} \]
Figure (svg): Two numbers in scientific notation divided by separating the parts
\[ 1.064 \times 10^8 \qquad 2.5 \times 10^{-4} \]
Verify: check the sizes are plausible
Why: The product of numbers around ten thousand and ten thousand should be around a hundred million, which is ten to the eighth. The quotient of ten by fifty thousand should be a small fraction, which negative four gives. Checking the order of magnitude catches a wrong exponent immediately.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.5 Scientific Notation §8.5, pp. 470-470
Faded example
Coefficients and powers separately.
Fill in the blanks
(1.4)(7.6) = 10.64, \; 10^4 \cdot 10^3 = 10^7} \;\Longrightarrow\; 10.64 \times 10^7 = 1.064 \times 10^8}
Why: The exponents add to seven, and moving the coefficient's point one place left raises the exponent to eight. The adjustment leaves the value unchanged and puts the answer into the required form.
Worked example
This is Example 3, part c.
\[ \text{Compute } \; (4 \times 10^2)^3. \]
Distribute the exponent
Why: The power of a product rule.
\[ 4 ^{3} x(10 ^{2}) ^{3} \]
Apply the power of a power rule
Why: Two times three is six.
\[ 64 x 10 ^{6} \]
Adjust
Why: Sixty-four is too large, so move left once.
\[ 6.4 x 10 ^{1} x 10 ^{6} \]
Combine the powers
Why: One plus six.
\[ 6.4 x 10 ^{7} \]
Figure (svg): A result adjusted back into scientific notation in both directions
\[ 6.4 \times 10^7 \]
Verify: check by rough size
Why: Four hundred cubed is sixty-four million, which is 6.4 times ten to the seventh. Doing the rough computation confirms both the coefficient and the exponent, and it takes less time than the formal route.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.5 Scientific Notation §8.5, pp. 470-470
Trap
\[ (1.4 \times 10^4)(7.6 \times 10^3) = 10.64 \times 10^7 \]
Report this as the answer, since the arithmetic is finished
Why: Both parts have been combined correctly and the value is right.
The coefficient of ten point six four is at least ten, so the answer is not in scientific notation. One more move left, with the exponent raised to eight, is needed.
\[ 10.64 \times 10^7 = 1.064 \times 10^8 \]
Check the coefficient's range as the final step
Why: The arithmetic often pushes it outside, and restoring the form is part of the answer.
The adjustment never changes the value, so it costs nothing but a line.
Sorting
Compare the coefficient with the allowed range.
Sort into buckets
Sort each intermediate result by the adjustment it needs.
The adjustment direction is decided entirely by the coefficient, and the exponent always moves the opposite way so that the value is preserved.
Elimination
Multiply 2.3 times ten cubed by 1.8 times ten to the fifth.
Eliminate the wrong options
Which is the answer in scientific notation?
Survives elimination: A
Why: The coefficients multiply to 4.14, which is already in range, and the exponents add to eight. No adjustment was needed here, which happens whenever the coefficients' product stays below ten.
Socratic
The four factors are rearranged freely.
Discussion prompt
Explain which properties permit the coefficients and the powers to be gathered separately when two such numbers are multiplied. Then say why the same regrouping works for division.
Hint: Think about Chapter 2's properties.
Answer:
Multiplication is commutative and associative, so the four factors of the product can be reordered and regrouped in any way. Putting the two coefficients together and the two powers together is one such regrouping, and it changes nothing about the value.
For division, writing the quotient as a product of two fractions is the same idea: the coefficients form one fraction and the powers another, and multiplying those two fractions reproduces the original. So one property covers both operations, which is why the two computations look so similar.
Section
Section 5
Concept
The exponent of a number in scientific notation gives its order of magnitude. Comparing exponents compares sizes immediately, which is often all a comparison needs.
This is why the notation is standard in science.
Figure (svg): Several quantities placed on a scale of powers of ten
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.5 Scientific Notation §8.5, pp. 469-474 — the lesson's use of the notation for very large and very small quantities
Picture it
Each step of three is a thousandfold.
Figure (svg): Several quantities placed on a scale of powers of ten
A scale of exponents fits quantities from a virus to a city on one line, which no ordinary number line could do. That compression is the notation's other main use.
Worked example
This is Example 5's situation in the textbook.
\[ \text{Alaska cost about } 7.2 \times 10^6 \text{ dollars and covers about } 5.9 \times 10^5 \text{ square miles.} \]
Write the quotient
Why: Price over area.
\[ \frac{7.2 x 10 ^{6}}{5.9 x 10 ^{5}} \]
Divide the coefficients
Why: Seven point two over five point nine is about 1.22.
\[ 1.22 \]
Subtract the exponents
Why: Six minus five.
\[ 10 ^{1} \]
Combine and interpret
Why: About twelve dollars per square mile.
\[ \text{about } \$ 12 \]
Figure (svg): A total divided by an area, both in scientific notation
\[ \approx 1.22 \times 10^1 = 12.2 \]
Verify: sanity-check the order of magnitude
Why: Seven million divided by six hundred thousand should be somewhere around twelve, since six hundred thousand goes into seven million about twelve times. The exponent arithmetic gave ten to the first, which is the right scale, so the answer is plausible before the coefficients are even divided precisely.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.5 Scientific Notation §8.5, pp. 474-474
Sorting
Compare exponents first.
Sort into buckets
Sort each pair by which number is larger.
The fourth pair is the close one: exponents differing by one beat coefficients differing by nine, since 1.1 times ten to the seventh is eleven million against nine point nine million.
Worked example
The exponents do most of the work.
\[ \text{How many times larger is } 6 \times 10^9 \text{ than } 3 \times 10^5? \]
Divide the coefficients
Why: Six over three is two.
\[ 2 \]
Subtract the exponents
Why: Nine minus five is four.
\[ 10 ^{4} \]
Combine
Why: Two times ten thousand.
\[ 20 000 \]
Interpret
Why: The first is twenty thousand times the second.
\[ 20 000 \times \]
Figure (svg): Several quantities placed on a scale of powers of ten
\[ 2 \times 10^4 = 20\,000 \]
Verify: check by rough magnitude alone
Why: The exponents differ by four, so the ratio is around ten thousand before the coefficients are considered, and the coefficients double it. Reading the exponents first gives the scale and the coefficients then refine it — which is how such comparisons are usually made.
Trap
\[ 3 \times 10^8 \quad \text{against} \quad 9 \times 10^5 \]
Say the second is larger, since nine exceeds three
Why: The coefficients are the visible numbers and nine is plainly bigger.
The exponents differ by three, which is a factor of a thousand, so the first is about three hundred times larger. The exponent dominates whenever the exponents differ at all.
Compare the exponents first and the coefficients only if the exponents match
Why: A difference of one in the exponent outweighs any difference in coefficients.
The coefficients are always between one and ten, so they can differ by at most a factor of ten — less than a single step in the exponent.
Faded example
Coefficients divide, exponents subtract.
Fill in the blanks
\dfrac24 = ___ \times 10^___} = 20\,000
Why: The exponents give the scale of ten thousand and the coefficients double it. Doing them separately is what makes comparisons of enormous numbers a matter of single-digit arithmetic.
Hypothesis
Predict before you decide.
Predict first
Can a difference in coefficients ever outweigh a difference of one in the exponents?
Correct: No, since coefficients differ by less than a factor of ten.
\[ 9.9 \times 10^6 < 1.1 \times 10^7 \]
Why: Every coefficient lies between one and ten, so the largest possible ratio between two coefficients is just under ten to one. A single step in the exponent is a factor of exactly ten, which is more. So the exponent always wins, and comparing exponents first is a reliable rule rather than a heuristic.
Socratic
Scientists could write the digits out.
Discussion prompt
Give two reasons scientific notation is used for measured quantities rather than plain decimal form. Then say what a scientist can tell from the exponent alone.
Hint: Think about size and about arithmetic.
Answer:
First, it makes enormous and tiny numbers readable: 0.000000000000000000000001673 is hard to take in, while 1.673 times ten to the negative twenty-four is not. Second, it turns multiplication and division of such numbers into small arithmetic on the coefficients plus addition or subtraction of the exponents.
From the exponent alone a scientist reads the order of magnitude, which is often the whole point of a comparison: whether a quantity is thousands or billions, whether two measurements are of comparable size, whether a calculation's answer is plausible. Getting the exponent right matters far more than getting the third digit of the coefficient right.
Comparison
Fill the blanks from memory before you scroll back.
Comparison matrix
| To decimal form | To scientific notation | |
|---|---|---|
| You are given | a coefficient and an exponent | a decimal number |
| Positive exponent means | move the point right | the number was at least 10 |
| Negative exponent means | move the point left | the number was below 1 |
The two directions are inverses, so converting back is always available as a check on a conversion.
Pattern
Whether converting or computing, the same five moves cover it.
Step five catches the errors that matter most. A wrong third digit is a small mistake and a wrong exponent is a factor of ten or more.
OpenStax Elementary Algebra 2e, §6.7 Integer Exponents and Scientific Notation §6.7
Check
The exponent counts the places.
Check your understanding
Write 5.6 times ten to the negative four in decimal form.
Answer: A
Why: A negative exponent moves the point four places left, giving three zeros between the point and the five. The number is below one, as the negative exponent requires.
Check
One nonzero digit before the point.
Check your understanding
Write 0.00062 in scientific notation.
Answer: A
Why: The point moves four places right to reach 6.2, and the number is below one so the exponent is negative four. Converting back confirms it.
Check
Combine, then adjust.
Check your understanding
What is (5 x 10^4) squared, in scientific notation?
Answer: A
Why: Five squared is twenty-five and the exponent doubles to eight, giving twenty-five times ten to the eighth. Adjusting the coefficient to 2.5 raises the exponent to nine.
Real world
This is Example 5's situation. In 1867 the United States bought Alaska for about 7.2 million dollars. Alaska covers about 590,000 square miles.
Discussion prompt
Write both figures in scientific notation, compute the price per square mile, and say what checking the order of magnitude tells you before the division is done.
Hint: Divide the coefficients and subtract the exponents.
Answer:
\[ \dfrac{7.2 \times 10^6}{5.9 \times 10^5} \approx 1.22 \times 10^1 \approx 12 \]
The price was about twelve dollars per square mile — famously cheap, and the arithmetic that shows it is a single-digit division plus a subtraction of exponents.
Before dividing the coefficients you already know the answer is around ten, because the exponents differ by one and both coefficients are between five and eight, so their ratio is near one. That order-of-magnitude check would catch an answer of a hundred and twenty or of one point two, which is the kind of error a misplaced decimal point produces.
Commit first
Answer, then rate your confidence honestly. Confident and wrong is the combination worth finding.
Predict first
Is 10.64 times ten to the seventh written in scientific notation?
Correct: No, because the coefficient must be less than 10.
\[ 10.64 \times 10^7 = 1.064 \times 10^8 \]
Why: The form requires the coefficient to be at least one and strictly less than ten, and 10.64 fails that. The value is correct, so the number is not wrong — only its written form is, and moving the point one place left with the exponent raised to eight fixes it. The restriction exists so that every number has exactly one representation, which is what makes two numbers comparable by inspection.
Explain it
They have seen the notation on a calculator and never used it deliberately.
Discussion prompt
In no more than four sentences, explain what scientific notation is and how to convert both ways. Then tell them the check that catches the commonest error.
Hint: One digit before the point, and the exponent counts places.
Answer:
A usable answer: you write every number as a single digit, a decimal part, and a power of ten — so thirty-four thousand becomes 3.4 times ten to the fourth. To convert one way you move the decimal point by the exponent, right for a positive one and left for a negative one, and to convert back you move the point until there is one digit in front and count the moves.
The check is to ask whether the number should be big or small before you start. A positive exponent means at least ten and a negative one means below one, so if your answer comes out on the wrong side of one you have moved the point the wrong way.
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: The sign is fixed by asking whether the number is large or small before converting. The coefficient's range is fixed by checking it as the final step of every problem. The adjustment is fixed by remembering that the exponent always moves opposite to the point. Comparisons are fixed by reading exponents first, since coefficients can never differ by a full factor of ten. 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 the form of scientific notation with the restriction on the coefficient, and beside it write three numbers that satisfy it and three that do not, correcting each of the three that fail. Underneath, convert four numbers to decimal form — two with positive exponents and two with negative — marking the direction and count of the point's movement on each. Then convert four decimal numbers the other way, checking each by converting it back. In the middle, multiply two numbers in scientific notation and divide two others, showing the regrouping, the combining and the final adjustment as separate lines, and circling the adjustment step. In the lower half, write two real quantities of very different sizes in scientific notation, compute their ratio, and write one sentence saying how many times larger one is. Finally, in the margin, write what the exponent's sign and size each tell you about a number.
Each of your conversions should return the original number when converted back. If one does not, count the places again — being one out is by far the most common conversion error.
Recap
Five things, and the second is where a factor of ten goes missing.
| If the question says | Your first move is |
|---|---|
| Write in decimal form | Move the point by the exponent |
| Write in scientific notation | Place the point after the first nonzero digit |
| Multiply two such numbers | Multiply coefficients, add exponents |
| The coefficient came out above 10 | Move the point left, raise the exponent |
| Which is larger | Compare the exponents first |
Lesson 8.6 returns to exponential functions with a purpose. A quantity growing by a fixed percentage each period is modelled by an exponential growth function, and the notation of this chapter is what makes such models manageable.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 8 Exponents and Exponential Functions — Lesson 8.5 Scientific Notation §8.5, pp. 469-474 — everything on these slides traces back here
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