Introduction

Exponents often show up in expressions with other operations. To evaluate these expressions, we follow the order of operations.

The order of the operations we perform follows the acronym PEMDAS:

  • First, we calculate any operations inside Parentheses, ().

  • Next, we evaluate the Exponents (a^b).

  • Then, we perform the Multiplications ( \times ) and Divisions ( \div ).

  • Finally, we complete any Additions ( + ) and Subtractions ( - ).

For example, let's look at the following expression: 3 \times 2^2

To evaluate this expression, we use PEMDAS. Here, we do not have Parentheses, so we move on to Exponents: 3\,\times\overbrace{2^2}^{\large\color{blue} 4} = 3\times \color{blue} 4

Then, we perform the Multiplication: 3\times 4 = 12

Therefore, 3 \times 2^2=12.

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Example: Evaluating Expressions With Stand-Alone Exponents

Evaluate 4^2 +(5-3).

EXPLANATION

To evaluate this expression, we use PEMDAS. We start with the Parentheses: 4^2\,+\underbrace{(5-3)}_{\large\color{blue} 2} =4^2\,+ {\color{blue} 2}

Then, we compute the Exponents: \overbrace{4^2}^{\large\color{red} 16}+2 = {\color{red} 16}+2

Finally, we perform the Addition: 16+2 = 18

Therefore, 4^2 +(5-3)=18.

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Practice: Evaluating Expressions With Stand-Alone Exponents

$3^2 \times 2 =$

a
 $36$
b
 $40$
c
 $81$
d
 $6$
e
 $18$
Practice: Evaluating Expressions With Stand-Alone Exponents

$(6-2) + 2^3 =$

a
 $8$
b
 $6$
c
 $16$
d
 $12$
e
 $20$
Practice: Evaluating Expressions With Stand-Alone Exponents

$3^2-2^3 = $

a
b
c
d
e
Example: Evaluating Expressions With Exponents Outside Parentheses

Calculate 28\div(7-5)^2.

EXPLANATION

To evaluate this expression, we use PEMDAS. We start with the Parentheses: 28\div \underbrace{(7-5)}_{\large\color{blue} 2}{}^2 =28\div {\color{blue} 2}^2

Then, we compute the Exponents: 28\div\overbrace{2^2}^{\large\color{red} 4} =28\div {\color{red} 4}

Finally, we perform the Division: 28\div 4 =7

Therefore, 28\div(7-5)^2=7.

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Practice: Evaluating Expressions With Exponents Outside Parentheses

$(2+1)^2=$

a
 $5$
b
 $16$
c
 $1$
d
 $4$
e
 $9$
Practice: Evaluating Expressions With Exponents Outside Parentheses

$5^2-(3+1)^2 = $

a
b
c
d
e
Practice: Evaluating Expressions With Exponents Outside Parentheses

$(8-2)^2\div 4 = $

a
b
c
d
e
Example: Evaluating Expressions With Exponents Inside Parentheses

Evaluate 5\times \left(3+2^2\right).

EXPLANATION

According to PEMDAS, we should first calculate the expression inside the Parentheses: \left(3+2^2\right).

To evaluate \left(3+2^2\right), we should again apply PEMDAS. We start with the Exponents: 3\,+\overbrace{2^2}^{\large\color{blue} 4} =3+ {\color{blue} 4}

Then, we perform the Addition: 3 + 4 = {\color{red} 7}

So we have: 5\times \underbrace{\left(3+2^2\right)}_{\large\color{red} 7} =5\times {\color{red} 7}

Now that we have calculated the expression inside the Parentheses, we can finally move on to the Multiplication: 5 \times 7 = 35

Therefore, 5\times\left(3+2^2\right)=35.

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Practice: Evaluating Expressions With Exponents Inside Parentheses

$80\div \left(3^3-7\right)=$

a
b
c
d
e
Practice: Evaluating Expressions With Exponents Inside Parentheses

$\left(3^3+3\right)\times 3=$

a
b
c
d
e
Practice: Evaluating Expressions With Exponents Inside Parentheses

$\left(3^2-5\right)\times 3=$

a
 $3$
b
 $9$
c
 $12$
d
 $4$
e
 $6$
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