Grade 5 · 5.NBT.A.2

Powers of ten

Ten ones make a ten, ten tens make a hundred, ten hundreds make a thousand. How can we write ten multiplied by itself many times, without counting a long row of zeros?

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Multiplying and dividing by ten, one hundred and one thousandGrade 4

Unlocks

Writing and reading scientific notationMath 8

Where you are 3 in Exponent notation

Number9

Exponents and roots5

Exponent notation3

Base, exponent and the value of a power Perfect squares and perfect cubes

Powers of ten

Why this exists

Fernanda is a bank teller. Loose dollar bills are slow to count, so the bank packs them. A strap holds 100 bills. Ten straps are packed into a brick, so a brick holds 1{,}000 bills. Each package is ten of the package before it.

Ten keeps showing up because the way we write numbers is built on it. Ten ones make a ten, ten tens make a hundred, ten hundreds make a thousand, and it never stops. Each place is worth ten times the place to its right, as in Multiplying and dividing by ten, one hundred and one thousand.

The trouble is that these numbers get long. A billion bills is written 1{,}000{,}000{,}000. To read it you have to count the zeros one at a time, and one missed zero makes the number ten times too small. This lesson gives each of these numbers a short name that says, right in the name, how many times ten was multiplied: a billion is 10^9.

Definition

A power of ten is the number we get by starting at 1 and multiplying by 10 some whole number of times. We write it as 10^n. The small raised number n counts the multiplications.

10^n = 1 \times \underbrace{10 \times 10 \times \cdots \times 10}_{n \text{ tens}}

The 10 is the base and n is the exponent, the same words used in Base, exponent and the value of a power. In this lesson n is always a whole number: 0, 1, 2, 3 and so on.

So 10^1 = 1 \times 10 = 10, and 10^2 = 1 \times 10 \times 10 = 100, and 10^3 = 1 \times 10 \times 10 \times 10 = 1{,}000.

We read 10^n as “ten to the n-th power”, or just “ten to the n-th”. So 10^5 is “ten to the fifth power”. Two of them have short names: 10^2 is “ten squared” and 10^3 is “ten cubed”.

When n = 0 we start at 1 and multiply by 10 zero times, so nothing changes:

10^0 = 1

That can look strange at first. The section “Why it is true” shows that it is the only value that fits the pattern of the other powers.

Intuition

Base-ten blocks show powers of ten as shapes. One small square is 1. Ten of them in a strip make 10. Ten strips side by side make a square of 100, which is one of Fernanda’s straps. Ten of those squares stacked up make a block of 1{,}000, which is a brick.

Figure 1Each block is ten of the block before it. The exponent counts how many times we multiplied by 10.

Every step to the right is “times 10”. The number of steps from the single square is the exponent: the strap is two steps along, so it is 10^2, and the brick is three steps along, so it is 10^3. The single square is zero steps along, which is why it sits under 10^0.

The same thing happens in a place value chart. The ones column is where we start. The tens column is one step to the left, the hundreds column two steps, the thousands column three steps. So the columns themselves are powers of ten.

Figure 2Every column of the chart is a power of ten, so 6{,}203 = 6 \times 10^3 + 2 \times 10^2 + 0 \times 10^1 + 3 \times 10^0.

The digit in a column says how many of that power the number has. The number 6{,}203 has 6 thousands, 2 hundreds, 0 tens and 3 ones. That is six of 10^3, two of 10^2, none of 10^1 and three of 10^0.

Results

The count of zeros. Written out in our number system, 10^n is a 1 followed by n zeros.

10^4 = 10{,}000 \qquad 10^6 = 1{,}000{,}000

The first ten powers. Each row is ten times the row above it.

power	value	name
10^0	1	one
10^1	10	ten
10^2	100	one hundred
10^3	1{,}000	one thousand
10^4	10{,}000	ten thousand
10^5	100{,}000	one hundred thousand
10^6	1{,}000{,}000	one million
10^7	10{,}000{,}000	ten million
10^8	100{,}000{,}000	one hundred million
10^9	1{,}000{,}000{,}000	one billion

Place values are powers of ten. The place n columns to the left of the ones place is worth 10^n. The ones place is 10^0, the tens place 10^1, the hundreds place 10^2, and so on.

Expanded form with powers of ten. A whole number is the sum of each digit times the power of ten for its place. For example:

\begin{aligned} 6{,}203 &= 6 \times 10^3 + 2 \times 10^2 \\ &\quad + 0 \times 10^1 + 3 \times 10^0 \end{aligned}

A term with the digit 0, such as 0 \times 10^1, is worth nothing and may be left out of the sum. The zero cannot be left out of the written number, though: 6{,}203 needs its 0 to hold the tens place.

Multiplying by a power of ten. To multiply a whole number by 10^n, write n zeros after it. So 52 \times 10^3 = 52{,}000.

Why it is true

The count of zeros. 10^1 = 10 is a 1 with one zero. Each further factor of 10 moves every digit one place to the left and puts a 0 in the ones place, as in Multiplying and dividing by ten, one hundred and one thousand. So each factor adds exactly one zero to the end. After n factors of 10 there are n zeros.

Place values. The ones place is worth 1. Each place is worth ten times the place to its right. So one step to the left is worth 1 \times 10, two steps 1 \times 10 \times 10, and n steps is 1 times n tens, which is 10^n.

Why 10^0 = 1. Read the table from the bottom up. Each step up takes away one factor of 10, so it divides by 10:

\begin{aligned} 10^3 &= 1{,}000 \\ 10^2 &= 1{,}000 \div 10 = 100 \\ 10^1 &= 100 \div 10 = 10 \\ 10^0 &= 10 \div 10 = 1 \end{aligned}

The next value after 10 in this list has to be 10 \div 10, which is 1. Any other value for 10^0 would break the pattern. It also matches the chart: the ones place is zero steps from itself, and it is worth 1.

Multiplying by 10^n. Multiplying by 10^n is multiplying by 10, then by 10 again, n times in all. The associative property says the factors may be grouped one at a time like this. Each time, the digits move one place left and a 0 fills the ones place, so n zeros appear.

Worked examples

The lesson is taught in two parts. Examples 1 to 3 belong to the first part: what 10^n means, how to read it, and the pattern of zeros. Examples 4 to 7 belong to the second part: place values as powers of ten, expanded form, and multiplying a whole number by a power of ten.

Example 1

Writing and reading powers of ten

(a) Write 10 \times 10 \times 10 \times 10 as a power of ten and find its value. (b) Write 10^7 as a number and say it aloud. (c) Find the value of 10^1.

Solution

(a) There are four factors of 10, so the exponent is 4: 10^4. Its value is a 1 followed by four zeros, 10^4 = 10{,}000. Check by multiplying: 10 \times 10 = 100, then 1{,}000, then 10{,}000.

(b) The exponent 7 says seven factors of 10, so the number is a 1 followed by seven zeros: 10^7 = 10{,}000{,}000. Say “ten to the seventh power”. The number is ten million.

Example 2

From a number to a power of ten

Write each number as a power of ten. (a) 100{,}000 (b) one billion (c) 1

Solution

(a) The number is a 1 followed by five zeros. Each zero is one factor of 10, so 100{,}000 = 10^5.

(b) One billion is written 1{,}000{,}000{,}000. Count the zeros in groups of three: three groups of three is nine zeros. So one billion is 10^9.

Example 3

Packing the vault

Daichi works in the bank vault with Fernanda. A brick holds 10^3 bills. Ten bricks fit in a box, and ten boxes fit on a cart. How many bills are on a full cart? Write the answer as a power of ten and as a number. How many straps of 100 bills is that?

Solution

A brick is 10^3 bills. A box is ten bricks, so it has one more factor of 10: 10^4 bills. A cart is ten boxes, one more factor again: 10^5 bills. As a number, a cart holds 100{,}000 bills.

For the straps, count steps instead of bills. A strap is 10^2 bills. Going from a strap to a brick, a box and a cart takes three steps, each “times 10”. So a cart holds 10^3 = 1{,}000 straps. Check: 1{,}000 straps of 100 bills is 1{,}000 \times 100 = 100{,}000 bills.

Example 4

Expanded form with powers of ten

Write 58{,}047 in expanded form using powers of ten.

Solution

Put the digits in a place value chart. The 5 is in the ten thousands place, four steps left of the ones, so it stands for 5 \times 10^4. The 8 is in the thousands place, three steps left: 8 \times 10^3. The 0 is in the hundreds place: 0 \times 10^2. The 4 is in the tens place: 4 \times 10^1. The 7 is in the ones place: 7 \times 10^0.

\begin{aligned} 58{,}047 &= 5 \times 10^4 + 8 \times 10^3 \\ &\quad + 0 \times 10^2 + 4 \times 10^1 \\ &\quad + 7 \times 10^0 \end{aligned}

Check by working out each term: 50{,}000 + 8{,}000 + 0 + 40 + 7 = 58{,}047. The 0 \times 10^2 term could be left out of the sum, since it is worth 0.

Example 5

From powers back to a number

Write as a single number: 9 \times 10^1 + 4 \times 10^5 + 2 \times 10^0 + 7 \times 10^3

Solution

The terms are out of order. The commutative property of addition lets us reorder them, largest power first:

\begin{aligned} &4 \times 10^5 + 7 \times 10^3 \\ &\quad + 9 \times 10^1 + 2 \times 10^0 \end{aligned}

The largest power is 10^5, so the number has six places, from 10^5 down to 10^0. Fill each place with its digit, and put a 0 in each place that has no term:

10^5	10^4	10^3	10^2	10^1	10^0
4	0	7	0	9	2

The number is 407{,}092. Check: 400{,}000 + 7{,}000 + 90 + 2 = 407{,}092.

Example 6

Counting straps and bricks

Kabir counts a stack of straps and a stack of bricks. Find each number of bills. (a) 36 straps, which is 36 \times 10^2 bills (b) 36 bricks, which is 36 \times 10^3 bills (c) 250 \times 10^4

Solution

(a) 10^2 is 10 \times 10. The associative property lets us multiply by one 10 at a time: 36 \times 10 = 360, then 360 \times 10 = 3{,}600. Each factor of 10 wrote one zero, so 36 \times 10^2 = 3{,}600 bills.

(b) Three factors of 10 write three zeros: 36 \times 10^3 = 36{,}000 bills. A brick is ten times a strap, so this is ten times the answer to (a), as it should be.

(c) Write four zeros after 250: 250 \times 10^4 = 2{,}500{,}000. The answer ends in five zeros, not four. The 0 already in 250 stays, and the four new ones go after it.

Example 7

Which vault holds more?

Daichi’s vault holds 7 \times 10^5 bills. Kabir’s vault holds 65 \times 10^4 bills. Kabir says his holds more because 65 is bigger than 7. Which vault holds more, and by how many bills?

Solution

The two numbers use different powers of ten, so 65 and 7 count different things: 65 ten thousands against 7 hundred thousands. Write both as numbers:

\begin{aligned} 7 \times 10^5 &= 700{,}000 \\ 65 \times 10^4 &= 650{,}000 \end{aligned}

Daichi’s vault holds more. The difference is 700{,}000 - 650{,}000 = 50{,}000 bills, which is 5 \times 10^4.

Expanded form shows the same thing without working out the numbers. Ten of 10^4 make 10^5, so 65 \times 10^4 is 6 \times 10^5 + 5 \times 10^4. Kabir has 6 hundred thousands and Daichi has 7, and the larger place decides which number is bigger.

Boundaries

Whole-number exponents only. Here n is 0, 1, 2 and so on. Exponents below zero exist too, and they give tenths, hundredths and smaller parts, but they belong to a later course. The decimal places themselves are in Thousandths: reading, writing and expanded form.

10^0 = 1 is a definition that fits the pattern. It is not “ten multiplied zero times gives zero”. Starting at 1 and doing no multiplying leaves 1.

The zeros rule is about our base-ten way of writing numbers. “10^n is a 1 with n zeros” works because our places are tens. In other number bases the places are powers of a different number.

Writing zeros after a number works for whole numbers. For a decimal such as 3.6, putting a 0 at the end does not change the value: 3.60 = 3.6. Multiplying or dividing a decimal by a power of ten moves the digits past the decimal point instead, which is taught in Multiplying a decimal by ten, one hundred or one thousand and Dividing a decimal by ten, one hundred or one thousand.

A zero term may leave the sum, never the number. In expanded form, 0 \times 10^1 can be dropped. In the written number, the 0 must stay to keep every other digit in its place.

Where this goes next

The same chart keeps going to the right of the ones place. Each step right divides by 10, which gives tenths, hundredths and Thousandths: reading, writing and expanded form. Moving digits across the decimal point is taught in Multiplying a decimal by ten, one hundred or one thousand and Dividing a decimal by ten, one hundred or one thousand.

Powers of ten are the base of Scientific notation. A very large number such as the number of stars in a galaxy is written as a small number times a power of ten, which is Writing and reading scientific notation.

Ten is not the only base. Base, exponent and the value of a power uses the same notation for any number multiplied by itself, such as 3^4. Counting factors also shows that 10^2 \times 10^3 = 10^5, since two tens and three more tens make five tens; that rule for any base is Multiplying and dividing powers with the same base.

Exercises

Part 1: writing and reading powers of ten

Write 10 \times 10 \times 10 \times 10 \times 10 \times 10 as a power of ten and find its value.
Write 10^8 as a number and say it aloud. What is the number called?
Find the values of 10^2 and 10^3, and give the short name for each power.
Write each number as a power of ten: (a) 1{,}000 (b) 10{,}000{,}000 (c) 10
How many zeros follow the 1 when 10^9 is written out? Write the number and name it.
Which is larger, 10^5 or 10^4? How many times as large?
Start from 10^5 = 100{,}000. Write the values of 10^4, 10^3, 10^2, 10^1 and 10^0, and say what each step down does.
Fernanda has a stack of 10^4 bills. How many straps of 100 is that? How many bricks of 1{,}000? Write each answer as a power of ten.
Write as powers of ten: (a) one hundred thousand (b) ten thousand (c) one hundred
Find the missing exponent: (a) 10^{?} = 1{,}000{,}000 (b) 10^{?} = 1
Put in order from smallest to largest: 10^3, 100, 10^5, 10{,}000, 10^0.

Part 2: place value, expanded form and multiplying

Write 4{,}815 in expanded form using powers of ten.
Write 30{,}506 in expanded form using powers of ten, with a term for every place.
Write 72{,}000{,}000 in expanded form using powers of ten, leaving out the zero terms.
Write as a number: 5 \times 10^4 + 3 \times 10^3 + 8 \times 10^1 + 1 \times 10^0
Write as a number: 2 \times 10^6 + 9 \times 10^2
Write as a number: 6 \times 10^0 + 4 \times 10^2 + 1 \times 10^5
Find (a) 47 \times 10^2 and (b) 47 \times 10^5.
Find 380 \times 10^3.
Kabir counts 23 bricks and 6 straps. How many bills is that?
Daichi needs 54{,}000 bills. How many bricks is that? If he uses only straps, how many straps?
Which is larger? (a) 9 \times 10^4 or 12 \times 10^3 (b) 3 \times 10^6 or 29 \times 10^5

Reasoning

Explain why 10^0 = 1, using the values of 10^3, 10^2 and 10^1.
In 5{,}555 every digit is a 5. Use powers of ten to explain why the four 5s are worth different amounts. How many times as much is the leftmost 5 worth as the rightmost 5?
Fernanda says that to multiply any whole number by 10^4 you write four zeros after it. Explain why she is right. Then explain why 380 \times 10^4 ends in five zeros, not four.

Answers

10^6 = 1{,}000{,}000. There are six factors of 10, so the exponent is 6, and the value is a 1 followed by six zeros.
10^8 = 100{,}000{,}000, said “ten to the eighth power”. The exponent gives eight zeros after the 1. The number is one hundred million.
10^2 = 100, “ten squared”; 10^3 = 1{,}000, “ten cubed”. Two zeros and three zeros.
(a) 1{,}000 = 10^3 (b) 10{,}000{,}000 = 10^7 (c) 10 = 10^1. Count the zeros after the 1.
Nine zeros: 10^9 = 1{,}000{,}000{,}000, one billion.
10^5 is larger: 10^5 = 100{,}000 and 10^4 = 10{,}000. It has one more factor of 10, so it is 10 times as large.
10^4 = 10{,}000, 10^3 = 1{,}000, 10^2 = 100, 10^1 = 10, 10^0 = 1. Each step down takes away one factor of 10, so it divides by 10.
10^2 straps and 10^1 bricks. 10^4 = 10{,}000 bills; 10{,}000 \div 100 = 100 = 10^2 straps, and 10{,}000 \div 1{,}000 = 10 = 10^1 bricks. Counted in steps: a strap is two steps below 10^4, a brick one step.
(a) 10^5 (b) 10^4 (c) 10^2. One hundred thousand is 100{,}000, five zeros; ten thousand is 10{,}000, four zeros; one hundred is 100, two zeros.
(a) 6, since 1{,}000{,}000 has six zeros. (b) 0, since 10^0 = 1.
10^0, 100, 10^3, 10{,}000, 10^5. As numbers these are 1, 100, 1{,}000, 10{,}000 and 100{,}000.
4{,}815 = 4 \times 10^3 + 8 \times 10^2 + 1 \times 10^1 + 5 \times 10^0. Check: 4{,}000 + 800 + 10 + 5 = 4{,}815.
30{,}506 = 3 \times 10^4 + 0 \times 10^3 + 5 \times 10^2 + 0 \times 10^1 + 6 \times 10^0. The 3 is in the ten thousands place, four steps left of the ones.
72{,}000{,}000 = 7 \times 10^7 + 2 \times 10^6. The 7 is seven places left of the ones and the 2 six places; every other digit is 0.
53{,}081. The places from 10^4 down hold 5, 3, 0, 8, 1; the 10^2 place has no term, so it gets a 0.
2{,}000{,}900. That is 2{,}000{,}000 + 900: a 2 in the 10^6 place, a 9 in the 10^2 place and zeros in the other five places.
100{,}406. Reorder by the commutative property: 1 \times 10^5 + 4 \times 10^2 + 6 \times 10^0, which is 100{,}000 + 400 + 6.
(a) 47 \times 10^2 = 4{,}700 (b) 47 \times 10^5 = 4{,}700{,}000. Write two zeros, then five zeros, after 47.
380 \times 10^3 = 380{,}000. Three new zeros go after the 0 already in 380.
23{,}600 bills. 23 \times 10^3 + 6 \times 10^2 = 23{,}000 + 600 = 23{,}600.
54 bricks, or 540 straps. 54{,}000 = 54 \times 10^3, and also 54{,}000 = 540 \times 10^2.
(a) 9 \times 10^4 is larger: 90{,}000 against 12{,}000. (b) 3 \times 10^6 is larger: 3{,}000{,}000 against 2{,}900{,}000, since 29 \times 10^5 = 2 \times 10^6 + 9 \times 10^5.
10^3 = 1{,}000, 10^2 = 100 and 10^1 = 10: each time the exponent goes down by one, a factor of 10 is taken away, so the value is divided by 10. The next step gives 10^0 = 10 \div 10 = 1. Any other value would break the pattern. It also matches the meaning: start at 1 and multiply by 10 zero times.
5{,}555 = 5 \times 10^3 + 5 \times 10^2 + 5 \times 10^1 + 5 \times 10^0. Each 5 is multiplied by the power of ten for its place, so the four 5s stand for 5{,}000, 500, 50 and 5. The leftmost is worth 5 \times 10^3 and the rightmost 5 \times 10^0, so it is worth 10^3, which is 1{,}000 times as much.
10^4 = 10 \times 10 \times 10 \times 10, so multiplying by 10^4 is multiplying by 10 four times (the associative property lets us do one factor at a time). Each time, every digit moves one place left and a 0 fills the ones place, so four zeros are written after the number. For 380 those four zeros go after the 0 that 380 already ends in, so 380 \times 10^4 = 3{,}800{,}000 ends in five zeros.