5th Grade · 5.NBT.A.2

Multiplying by powers of 10 in 5th grade

Multiplying by powers of 10 means multiplying by 10, 100, 1,000 and so on. In 5th grade, your child learns to write these numbers with exponents, such as $10^3$ for 1,000. Your child also learns the pattern behind the zeros and the decimal point when multiplying or dividing by 10, 100 or 1,000.

What the standard says

The official Common Core wording, quoted exactly. The plain version follows.

5.NBT.A.2 Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.

What it means, in plain words

Powers of 10 are the numbers you get when you multiply 10 by itself: 10, 100, 1,000 and so on. An exponent is the small raised number that says how many 10s are multiplied together. So $10^2 = 10 \times 10 = 100$ and $10^4 = 10 \times 10 \times 10 \times 10 = 10{,}000$ . For a power of 10, the exponent also counts the zeros. In 5th grade, children use exponents for powers of 10 only.

When your child multiplies a whole number by a power of 10, the product gets one zero at the end for each 10. For example, $45 \times 10^3 = 45{,}000$ . The reason is place value. Multiplying by 10 makes every digit worth 10 times as much, so each digit moves one place to the left.

The rule for multiplying by powers of 10 is the same for decimals. Each digit moves one place to the left, so the decimal point seems to move one place to the right: $0.6 \times 10 = 6$ . Dividing by a power of 10 does the opposite. The digits move to the right and the number gets smaller: $9.3 \div 10^2 = 0.093$ .

Your child should be able to:

Write 10, 100, 1,000 and 10,000 as powers of 10, such as $1{,}000 = 10^3$
Multiply a whole number by a power of 10 and explain the zeros in the product
Move the decimal point the right number of places when multiplying or dividing by 10, 100 or 1,000
Say whether the answer will be bigger or smaller before working it out

Example

What is $2.75 \times 10^2$ ?

$10^2 = 10 \times 10 = 100$ , so the problem is $2.75 \times 100$ .
Multiplying by 100 makes each digit 100 times as much, so each digit moves two places to the left.
The decimal point ends up two places to the right: 2.75 becomes 275.
So 2.75 times $10^2$ is 275.

Example

What is $58 \div 10^3$ ?

$10^3 = 1{,}000$ , so the problem is $58 \div 1{,}000$ .
Dividing by 1,000 moves each digit three places to the right.
The decimal point moves three places to the left. Empty places are filled with zeros: 58 becomes 0.058.
So 58 divided by $10^3$ is 0.058, which is fifty-eight thousandths.

Where children usually get stuck

Your child writes $2.6 \times 100 = 2.600$ .
With whole numbers, multiplying by 100 adds two zeros at the end. The child carries that shortcut over to decimals, where zeros at the end change nothing: 2.600 is still 2.6.
What helps. Ask whether the answer should be bigger or smaller than 2.6. Then put 2.6 in a place-value chart and slide each digit two places to the left. The answer is 260.
Your child writes $10^3 = 30$ .
The exponent gets read as "times 3". Nothing in the small raised number says to multiply 10 by itself.
What helps. Have your child write it out once: $10 \times 10 \times 10 = 1{,}000$ . Then count the zeros in 1,000 and compare the count with the exponent.
Your child writes $4.2 \div 10 = 42$ .
The child remembers to move the decimal point but not which way. Dividing by 10 must give a smaller number, and 42 is bigger than 4.2.
What helps. Use money. 4 dollars and 20 cents shared by 10 people is 42 cents each, or 0.42 dollars. Asking "bigger or smaller?" before moving the point catches this mistake.

Try four practice questions

Real 5th grade questions from our question bank, easiest first. Answer them right here, check each one, and see a worked solution for any question.

1Warm-up
Find the missing whole-number exponent that makes the equation true.
$\displaystyle 37 \times 10^{\text{?}} = 3{,}700$
Enter the missing exponent.
Exponent:
Show the answer and how to solve it
Answer. 2
The missing exponent is 2 because multiplying 37 by 10 twice gives 3,700.
1. 1Compare 37 and 3,700. The nonzero digits stay in the same order: 3, then 7.
2. 2Each time you multiply by 10, the digits move 1 place to the left.
3. 3The product matches when 37 is multiplied by 10 to the 2nd power, so the missing exponent is 2.
Common mistake. A common mistake is counting the two zeros in 3,700 and answering 2, which happens to be correct here, but students who then apply that same shortcut to numbers like 204 will be led astray. Focus on place-value shifts rather than counting zeros.
The idea underneath. A power of 10 does not change the order of the nonzero digits; it tells how many place-value positions the whole number shifts.
2On level
The equation below shows the same number in standard form and as a power of 10.
$\displaystyle 1 = 10^{\text{?}}$
What is the missing exponent?
Exponent:
Show the answer and how to solve it
Answer. 0
The missing exponent is 0 because 1 is $10^{0}$ , and 1 has zero zeros after it.
1. 1Read the whole number in the equation. It is 1.
2. 2For whole-number powers of 10, the exponent tells how many zeros come after the 1. The number 1 has no zeros after it, so the exponent is 0. This is a special case: $10^{0} = 1$ .
3. 3So the missing exponent is 0.
Common mistake. A common mistake is typing 1 instead of 0. The number 1 equals $10^{0}$ , not $10^{1}$ . Because 1 has no zeros after the digit 1, the exponent is 0.
The idea underneath. An exponent greater than 0 tells how many 10s to multiply together. For example, $10^2 = 10 \times 10 = 100$ . So the exponent also matches the number of zeros after the 1. The special case is $10^0 = 1$ .
3On level
A power of 10 is shown below.
$\displaystyle 10^{0}$
Write its value in the place-value chart.
Ten Thousands
Thousands
Hundreds
Tens
Ones
Show the answer and how to solve it
Answer. 1
The value is 1, so the chart shows 1 in Ones and all other columns blank.
1. 1Start with the power shown. The exponent is 0.
2. 2Any nonzero number raised to the power of 0 equals 1, so $10^{0} = 1$ .
3. 3The value 1 has only a digit in the Ones place. All other columns — Tens, Hundreds, Thousands, and Ten Thousands — are empty.
4. 4So the chart should have 1 in Ones and all other columns left blank.
Common mistake. A common mistake is putting 1 in the Tens column instead of Ones, or writing a 0 in other columns when they should stay blank. The exponent 0 means there are no zeros after the 1, so the only digit entered is 1 in Ones.
The idea underneath. For a power of 10, the exponent tells how many zeros follow the 1 and which place-value column gets the 1. When the exponent is 0, there are no zeros — the value is simply 1 in the Ones column.
4Stretch
Find the product shown below.
$\displaystyle 3040 \times 10^{3}$
Type the product.
Show the answer and how to solve it
Answer. 3040000
The answer is 3040000 because $10^{3} = 1000$ , and $3040 \times 1000 = 3040000$ .
1. 1Start with the power of 10 in the equation.
2. 2Multiplying by 1000 moves every digit in 3040 three places to the left. The zero that is already in 3040 stays in the number, and three new zeros are appended to the right.
3. 3So the product is 3040000.
Common mistake. A common mistake is writing 304000 or 30400000. The exponent 3 means multiply by 1000, so exactly three new zeros are appended. The trailing zero already in 3040 is preserved, giving four total zeros at the end of the product.
The idea underneath. Multiplying by 10 to the nth power changes place value, not digit order, so each digit moves n places to the left.