# Order of Operations: Exponents

šPractice powers and roots

As part of combined operations, we learned that parentheses always come first.

Once solved, we can begin to simplify powers (or roots).

After simplifying them, we can continue solving the exercise according to the order of basic operations:

Firstly, the multiplications and divisions, and lastly, the additions and subtractions.

Let's refresh the order of operations:

## Test yourself on powers and roots!

Which of the following is equivalent to $$100^0$$?

In the event that we have an exercise with parentheses that are within other parentheses, we will first solve the inner parentheses and then move on to the outer parentheses.

## Let's take a look at the following exercise with combined operations and see it step by step.

### For example, Exercise 1

$2 + 5 \cdot 4^2 \cdot (3-1) =$

First, we perform the operations inside the parentheses. Once done, we obtain the following:
$2 + 5 \cdot 4^2 \cdot 2 =$

Note that we have a combined operation involving powers, multiplications, and additions, so we proceed to solve the power.

Once done, we obtain:
$2+5 \cdot16 \cdot 2 =$

Now it's time to solve the multiplications (remember: from left to right):
$āā2+80\cdot2=$,$2+160=āā$

$2+160=162$

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### Exercise 2

Now we will do the same exercise, but with a small variation.

$2 + 5 \cdot 4^2 \cdot (3^2-1) =$

First, we must solve the operation inside the parentheses, where there is a power and a subtraction, so following the order of operations we calculate the power and then the subtraction.

$2 + 5 \cdot 4^2 \cdot (3^2-1) =$

$2 + 5 \cdot 4^2 \cdot (9-1) =$

Now we can proceed with solving the exercise just as we have been doing so far.

$2 + 5 \cdot 4^2 \cdot 8 =$

$2 + 5 \cdot 16 \cdot 8 =$

$2 + 80 \cdot 8 =$

$2 + 640 = 642$

### Exercise 3

$(23+4^2)+8\cdot(9^2-1)=$

We solve the operations within each parenthesis, applying the order of operations within them.

$(23+16)+8\cdot(81-1)=$

Now is the time to solve the multiplications (we remember: from left to right):

$39+8\cdot80=$

$39+640=$

$39+640=679$

Remember that the order of operations will always be the same, even when combined operations with fractions appear.

Do you know what the answer is?

## Combined Operations Exercises

### Basic Example 4

$4+2^2=$

First, the powers

$4+4=$

$4+4=8$

### Basic Example 5

$4+2+5^2=$

First, the powers

$4+2+25=$

$4+2+25=31$

### Basic Example 6

$5+5-5^2+4^2=$

First, the powers

$5+5-25+16=$

$5+5-25+16=1$

## Exercises to Practice the Order of Basic Operations (Powers)

• $3 \cdot3+3^2=$
• $(3+1)^2-(4+1)=$
• $10:2-2^2=$
• $100:5^2+3^2=$
• $5^3:5^2\cdot2^3=$
• $0:2^2\cdot1^{10}+3$
• $({1\over4})^2+{1\over16}=$
• $({1\over2})^2+({1\over3})^2+{1\over4}=$
• $8-3^2:3=$
• $(2+1 \cdot2)^2=$
• $(20-3 \cdot2^2)^2=$
• $(15+9:3-4^2)^2=$
• $[(4-2^2)]^3=$
• $5+8^2=$
• $2^6+3=$
• $12-3^2=$
• $22-3^4=$
• $({1\over3})^2\cdot60=$
• $7-8 \cdot 2-3^2=$
• $25\cdot[({1\over2})^2+2^2]=$
• $27.5+1.5^3\cdot6=$
• $0.2^2\cdot5=$
• $(6-6)\cdot2^2=$
• $1+20^2\cdot{1\over5}=$

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Do you think you will be able to solve it?

## Review Questions

### What is the order of operations when there are exponents and powers?

Exponents and roots should always be performed before multiplication or division; and before addition or subtraction.

### What is the correct order of mathematical operations?

When working with combined operations that include parentheses, exponents and roots, multiplications and divisions, as well as additions and subtractions, we must use the order of operations which indicates the sequence in which we should perform the operations.

• Parentheses are solved first. If there are parentheses within other parentheses, we solve the inner ones first and then the outer ones.
• Exponents and roots are solved next.
• Multiplications and divisions are solved (from left to right).
• Additions and subtractions are solved (from left to right).

### What Are Operations with Powers?

Powers are used to abbreviate the multiplication of a number (called the base) by itself, a certain number ānā of times.

When we have combined basic operations that include powers, we must remember that powers are to be resolved after the parentheses.

### How to Solve Order of Operations with Exponents?

We solve the parentheses, and subsequently we raise the bases to the indicated exponent, this is done by multiplying it by itself, as many times as the exponent indicates.

Do you know what the answer is?

## Examples with solutions for Order of Operations: (Exponents)

### Exercise #1

$5^3:5^2\times2^3=$

### Step-by-Step Solution

In the first stage, let's calculate the powers of each of the terms:

$5^3=5\times5\times5=25\times5=125$

$5^2=5\times5=25$

$2^3=2\times2\times2=4\times2=8$

Now let's write the resulting expression:

$125:25\times8=$

Since the only operations in the expression are multiplication and division, we will solve the expression from left to right

In other words, we will divide first and then multiply:

$125:25=5$

$5\times8=40$

40

### Exercise #2

Sovle:

$3^2+3^3$

### Step-by-Step Solution

Remember that according to the order of operations, exponents precede multiplication and division, which precede addition and subtraction (and parentheses always precede everything).

So first calculate the values of the terms in the power and then subtract between the results:

$3^2+3^3 =9+27=36$Therefore, the correct answer is option B.

36

### Exercise #3

Solve:

$5^2\cdot4+3^3$

### Step-by-Step Solution

Remember that according to the order of arithmetic operations, exponents precede multiplication and division, which precede addition and subtraction (and parentheses always precede everything).

So first calculate the values of the terms with exponents and then subtract the results:

$5^2\cdot4+3^3 =25\cdot4+27=100+27=127$Therefore, the correct answer is option B.

127

### Exercise #4

What is the answer to the following?

$3^2-3^3$

### Step-by-Step Solution

Remember that according to the order of operations, exponents come before multiplication and division, which come before addition and subtraction (and parentheses always before everything),

So first calculate the values of the terms in the power and then subtract between the results:

$3^2-3^3 =9-27=-18$Therefore, the correct answer is option A.

$-18$

### Exercise #5

Solve:

$\sqrt{4}\cdot4^2-5^2\cdot\sqrt{1}$

### Step-by-Step Solution

We simplify each term according to the order from left to right:

$\sqrt{4}=2$

$4^2=4\times4=16$

$5^2=5\times5=25$

$\sqrt{1}=1$

Now we rearrange the exercise accordingly:

$2\times16-25\times1$

Since there are two multiplication operations in the exercise, according to the order of operations we start with them and then subtract.

We put the two multiplication exercises in parentheses to avoid confusion during the solution, and solve from left to right:

$(2\times16)-(25\times1)=32-25=7$