Exponents: Repeated Factors and Powers of Ten | Grade 6 Math | LevelKite

You already know that factors are the numbers you multiply to get a product. In $3 \times 8 = 24$ , the factors are 3 and 8.

Sometimes one factor is used again and again, as in $2 \times 2 \times 2 \times 2 \times 2$ . A long product like this is slow to write and easy to miscount. In this lesson, you will learn a short way to write it. You will also learn how to find its value.

Here is the short way to write that product:

\underbrace{2 \times 2 \times 2 \times 2 \times 2}_{5 \text{ factors of } 2} = 2^{5}

The short form $2^5$ is called a power. A power has two parts:

The base is the number that is used as a factor. In $2^5$ , the base is 2.
The exponent is the small raised number. It tells how many times the base is used as a factor. In $2^5$ , the exponent is 5.

The exponent is a count of factors, so it is a whole number. Every exponent in this unit is a whole number.

You read $2^5$ as two to the fifth power. It names the same product as $2 \times 2 \times 2 \times 2 \times 2$ , so the two expressions are equal. Writing the power does not change the value. It only shortens the writing.

Key concept

The exponent counts factors of the base

A power is a short way to show how many times one number is used as a factor. The base is that number. The exponent tells how many times the base is used as a factor.

In the formula below, $b$ stands for the base and $n$ stands for the exponent. This meaning holds when $n$ is a whole number, 1 or greater. In Grade 6, every exponent is a whole number. In this lesson, the base is a whole number too. Later, the base may be a fraction or a decimal.

b^{n} = \underbrace{b \times b \times \cdots \times b}_{n \text{ factors of } b}

The same product shown as repeated multiplication, as a power, and in words

Repeated multiplication	Power	Base	Exponent	How to read it
$4 \times 4$	$4^2$	4	2	four to the second power, or four squared
$5 \times 5 \times 5$	$5^3$	5	3	five to the third power, or five cubed
$9 \times 9 \times 9 \times 9$	$9^4$	9	4	nine to the fourth power
$2 \times 2 \times 2 \times 2 \times 2$	$2^5$	2	5	two to the fifth power

Worked example

Write a product as a power

Write $7 \times 7 \times 7 \times 7$ using an exponent.

Find the number that is used as a factor. Every factor in this product is 7, so the base is 7.
$\text{base} = 7$
Count how many times 7 is used as a factor. There are 4 factors, so the exponent is the whole number 4.
$\underbrace{7 \times 7 \times 7 \times 7}_{4 \text{ factors of } 7}$
Write the base. Then write the exponent small and raised, to the right of the base. The power names the same product as the long form, so the two expressions are equal.
$7 \times 7 \times 7 \times 7 = 7^{4}$

Answer

Written with an exponent, $7 \times 7 \times 7 \times 7$ is $7^4$ .

To evaluate a power means to find its value. When the exponent is a whole number, 1 or greater, the meaning of a power gives you a method that always works.

First, write the base as a factor as many times as the exponent says. Then multiply from left to right, as you do in the order of operations. Keep track of the product so far, and bring in one more factor each time.

Before you multiply, count your factors. The count must match the exponent.

Worked example

Evaluate a power

Evaluate $3^4$ .

Name the parts. The base is 3 and the exponent is 4. The exponent 4 is a whole number, so it counts factors: write 3 as a factor 4 times.
$3^4 = 3 \times 3 \times 3 \times 3$
Multiply the first two factors, because you multiply from left to right.
$3 \times 3 = 9$
Multiply the product so far, 9, by the third factor of 3.
$9 \times 3 = 27$
Multiply the product so far, 27, by the fourth factor of 3. All 4 factors are now used, so this product is the value of the power.
$27 \times 3 = 81$

Answer

The value of $3^4$ is $81$ .

Worked example

Base and exponent have different jobs

Evaluate $2^5$ and $5^2$ . Do they have the same value?

In the first power, the base is 2 and the exponent is 5. The exponent is a whole number, so it counts factors: write 2 as a factor 5 times.
$2^5 = 2 \times 2 \times 2 \times 2 \times 2$
Multiply from left to right. The product so far is 4, then 8, then 16, then 32.
$2 \times 2 = 4,\quad 4 \times 2 = 8,\quad 8 \times 2 = 16,\quad 16 \times 2 = 32$
In the second power, the jobs are swapped. The base is 5 and the whole-number exponent is 2, so write 5 as a factor 2 times and multiply.
$5^2 = 5 \times 5 = 25$
Compare the two values. They are different numbers.
$32 \neq 25$

Answer

No. $2^5 = 32$ and $5^2 = 25$ , so these two powers do not have the same value. Swapping the base and the exponent can change the value, so check which number is which.

A power of ten is a power whose base is 10, such as $10^3$ . Powers of ten follow a pattern that comes from place value.

When you multiply a whole number by 10, each digit moves one place to the left, and a 0 fills the ones place. So $10 \times 10 = 100$ , and $100 \times 10 = 1{,}000$ . Each extra factor of 10 puts one more zero on the end.

An exponent of 1 means the base is used as a factor just once, so $10^1 = 10$ . The table starts there. Each row after it uses one more factor of 10.

Powers of ten from the first power to the sixth power

Power of ten	Written as factors of 10	Value	Zeros in the value
$10^1$	$10$	$10$	1
$10^2$	$10 \times 10$	$100$	2
$10^3$	$10 \times 10 \times 10$	$1{,}000$	3
$10^4$	$10 \times 10 \times 10 \times 10$	$10{,}000$	4
$10^5$	$10 \times 10 \times 10 \times 10 \times 10$	$100{,}000$	5
$10^6$	$10 \times 10 \times 10 \times 10 \times 10 \times 10$	$1{,}000{,}000$	6

Compare the exponent in the first column with the number of zeros in the last column. They match in every row. This is why: $10^1 = 10$ has one zero, and each extra factor of 10 adds one more zero.

So you can use this pattern: when the exponent is a whole number, 1 or greater, a power of ten is a 1 followed by as many zeros as the exponent. For example, $10^4$ is a 1 followed by 4 zeros, which is $10{,}000$ .

You can also work backward. Suppose a whole number is a 1 followed by one or more zeros, with no other digits. Count the zeros. That count is the exponent. For example, $1{,}000$ has 3 zeros, so $1{,}000 = 10^3$ .

Use this pattern only when the base is 10. It does not fit other bases. For example, $2^3 = 8$ , and 8 has no zeros at all.

Worked example

Write and evaluate powers of ten

Part A: Write $10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10$ as a power of ten. Then evaluate it. Part B: Write $100{,}000{,}000$ as a power of ten.

Part A. Every factor is 10, so the base is 10. Count the factors. There are 7, so the exponent is the whole number 7.
$10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 = 10^{7}$
Evaluate with the zeros pattern. The base is 10 and the exponent 7 is a whole number, so the value is a 1 followed by 7 zeros.
$10^{7} = 10{,}000{,}000$
Check with the table. It shows that six factors of 10 make 1,000,000. One more factor of 10 moves each digit one place to the left and adds a zero. The results agree.
$1{,}000{,}000 \times 10 = 10{,}000{,}000$
Part B. This whole number is a 1 followed only by zeros, so the pattern works backward. Count the zeros. There are 8, so the exponent is the whole number 8.
$100{,}000{,}000 = 10^{8}$
Check by building on Part A. Eight factors of 10 are the seven factors from Part A times one more 10.
$10{,}000{,}000 \times 10 = 100{,}000{,}000$

Answer

Part A: the product is $10^7$ , and its value is $10{,}000{,}000$ . Part B: written as a power of ten, $100{,}000{,}000$ is $10^8$ .

Here is the idea to take with you. In a power, the base is the number used as a factor, and the whole-number exponent counts how many times it is used. That lets you write a long product such as $7 \times 7 \times 7 \times 7$ in the short form $7^4$ .

To evaluate a power, write out the factors and multiply from left to right. When the base is 10 and the exponent is a whole number, 1 or greater, the value is a 1 followed by that many zeros.

Next, you will evaluate powers whose bases are decimals or fractions, and you will write numerical expressions that use exponents. The exponent will still be a whole number that counts factors.