Introduction

There's a way to write repeated multiplications like 2\times 2\times 2 and 5\times 5 in a more compact way.

For the expression 2\times 2\times 2, we have that \color{blue}2 is multiplied by itself \color{red}3 times: \underbrace{{\color{blue}2} \times {\color{blue}2} \times {\color{blue}2}}_{\large\text{[math]{\color{red}3}[/math] times}}

To write this more compactly, we can use the exponent form {\color{blue}2}^{\color{red}3}, pronounced as "two to the power of three" or "two to the third power".

Here, the whole expression {\color{blue}2}^{\color{red}3} is a power. We call \color{blue}2 the base, and \color{red}3 the exponent.

Likewise, for the expression 5\times 5, we have that \color{blue}5 is multiplied by itself \color{red}2 times: \underbrace{{\color{blue}5} \times {\color{blue}5}}_{\large\text{[math]{\color{red}2}[/math] times}}

So, in exponent form, the base must be \color{blue}5 and the exponent must be {\color{red}2}. Therefore, {\color{blue}5} \times {\color{blue}5} = {\color{blue}5}^{\color{red}2}.

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Example: Writing Repeated Multiplication of One-Digit Numbers Using Exponents

What expression is equivalent to 8 \times 8 \times 8 \times 8?

EXPLANATION

In this question, \color{blue}8 is multiplied by itself \color{red}4 times: \underbrace{{\color{blue}8} \times {\color{blue}8} \times {\color{blue}8} \times {\color{blue}8}}_{\large\text{[math]{\color{red}4}[/math] times}}

So, in exponent form, the base must be \color{blue}8 and the exponent must be {\color{red}4}. Therefore, {\color{blue}8} \times {\color{blue}8} \times {\color{blue}8} \times {\color{blue}8} = {\color{blue}8}^{\color{red}4}.

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Practice: Writing Repeated Multiplication of One-Digit Numbers Using Exponents

$2 \times 2 \times 2$ is equivalent to

a
 $2^2$
b
 $6$
c
 $3^3$
d
 $2^3$
e
 $3^2$
Practice: Writing Repeated Multiplication of One-Digit Numbers Using Exponents

Express the following product in exponent form.

a
b
c
d
e
Example: Writing Repeated Multiplication of Larger Numbers Using Exponents

What is 18 \times 18 \times 18 written as a power?

EXPLANATION

In this question, \color{blue}18 is multiplied by itself \color{red}3 times: \underbrace{{\color{blue}18} \times {\color{blue}18} \times {\color{blue}18}}_{\large\text{[math]{\color{red}3}[/math] times}}.

So, in exponent form, the base must be \color{blue}18 and the exponent must be {\color{red}3}. Therefore, {{\color{blue}18} \times {\color{blue}18} \times {\color{blue}18}}= {\color{blue}18}^{\color{red}3}.

Watch out! Notice that {\color{blue}18}^{\color{red}3} means that the base is {\color{blue}18} , not 8.

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Practice: Writing Repeated Multiplication of Larger Numbers Using Exponents

Express the following product in exponent form.

a
b
c
d
e
Practice: Writing Repeated Multiplication of Larger Numbers Using Exponents

$123 \times 123 \times 123 \times123 \times 123 $ is equivalent to

a
 $123\times 5$
b
 $213^5$
c
 $23^5$
d
 $123^5$
e
 $123^4$
Example: Writing Exponents as Repeated Multiplication

Write down 6^5 using multiplication signs.

EXPLANATION

Here, {\color{blue}6}^{\color{red}5} is in exponent form. The base is \color{blue}6 and the exponent is {\color{red}5}.

This means that \color{blue}6 is multiplied by itself \color{red}5 times: \underbrace{{\color{blue}6} \times {\color{blue}6} \times {\color{blue}6} \times {\color{blue}6} \times {\color{blue}6}}_{\large\text{[math]{\color{red}5}[/math] times}}

Therefore, 6^5 = 6 \times 6 \times 6 \times 6 \times 6 .

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Practice: Writing Exponents as Repeated Multiplication

$7^6=$

a
 $ 7 \times 7 \times 7 \times 7 \times 7$
b
 $7 \times 6$
c
 $6 \times 6 \times 6 \times 6 \times 6 \times 6 \times 6$
d
 $7 \times 7 \times 7 \times 7$
e
 $7 \times 7 \times 7 \times 7 \times 7 \times 7$
Practice: Writing Exponents as Repeated Multiplication

$5^4=$

a
 $4 \times 4 \times 4 \times 4 \times 4$
b
 $4 \times 4 \times 4 \times 4$
c
 $5 \times 5 \times 5 \times 5 \times 5$
d
 $5 \times 5 \times 5 \times 5$
e
 $5 \times 5 \times 5$
Example: Writing Larger Exponents as Repeated Multiplication

What is 234^4 expressed using multiplication signs?

EXPLANATION

We notice that {\color{blue}234}^{\color{red}4} is in exponent form.

First, note that the base is NOT 4 . The base is NOT 34 either.

Instead, the base is the full number {\color{blue}234}, and the exponent is {\color{red}4}. This means that \color{blue}234 is multiplied by itself \color{red}4 times: \underbrace{{\color{blue}234} \times {\color{blue}234} \times {\color{blue}234} \times {\color{blue}234} }_{\large\text{[math]{\color{red}4}[/math] times}}

Therefore, 234^4 = 234 \times 234 \times 234 \times 234 .

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Practice: Writing Larger Exponents as Repeated Multiplication

$12^3=$

a
 $ 12 + 12 +12$
b
 $12\times 3$
c
 $ 12 \times 12 \times 12$
d
 $36$
e
 $12$
Practice: Writing Larger Exponents as Repeated Multiplication

${208}^2=$

a
 $208\times208$
b
 $208$
c
 $208+208$
d
 $208+2$
e
 $208\times 2$
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