Order of Operations - Exponents and Roots

The second step in the order of operations is - exponents and roots!

2 English Order of Operations Exponents

Immediately after dealing with parentheses, we move on to exponents and roots!
Pay attention - even within parentheses, it's very important to maintain the correct order of operations!
Exponent - multiply the base by itself the number of times shown in the exponent (the small number on the top right).
Root - half power - which positive number when multiplied by itself will give the number written under the root.

Suggested Topics to Practice in Advance

  1. The Order of Basic Operations: Addition, Subtraction, and Multiplication

Practice Powers and Roots

Examples with solutions for Powers and Roots

Exercise #1

1052:5= 10-5^2:5=

Step-by-Step Solution

First, compute the power: 52=25 5^2 = 25 .

Next, divide: 25÷5=5 25 \div 5 = 5 .

Finally, subtract: 105=5 10 - 5 = 5 .

Answer

5 5

Exercise #2

1542:2= 15-4^2:2=

Step-by-Step Solution

First, compute the power: 42=16 4^2 = 16 .

Next, divide: 16÷2=8 16 \div 2 = 8 .

Finally, subtract: 158=7 15 - 8 = 7 .

Answer

7 7

Exercise #3

2033:3= 20-3^3:3=

Step-by-Step Solution

First, compute the power: 33=27 3^3 = 27 .

Next, divide: 27÷3=9 27 \div 3 = 9 .

Finally, subtract: 209=11 20 - 9 = 11 .

Answer

11 11

Exercise #4

3×2+81= 3 \times 2 + \sqrt{81} =

Step-by-Step Solution

First, evaluate the square root: 81=9\sqrt{81}=9.

Then, follow the order of operations (PEMDAS/BODMAS):

1. Multiplication: 3×2=63 \times 2 = 6

2. Addition: 6+9=156 + 9 = 15

So, the correct answer is 15 15 .

Answer

15 15

Exercise #5

4+49×3= 4 + \sqrt{49} \times 3 =

Step-by-Step Solution

First, solve the square root: 49=7 \sqrt{49} = 7 .

Next, multiply 7 by 3: 7×3=21 7 \times 3 = 21 .

Finally, add 4 to 21: 4+21=25 4 + 21 = 25 .

Answer

25 25

Exercise #6

5216+2= 5^2 - \sqrt{16} + 2 =

Step-by-Step Solution

Start by calculating the power: 52=25 5^2 = 25 .

Then, calculate the square root: 16=4 \sqrt{16} = 4 .

Subtract 4 from 25: 254=21 25 - 4 = 21 .

Finally, add 2: 21+2=23 21 + 2 = 23 .

Answer

23 23

Exercise #7

63+5×22= 6 - 3 + 5 \times 2^2 =

Step-by-Step Solution

First, follow the order of operations (BODMAS/BIDMAS):

Step 1: Calculate the exponent:
22=42^2 = 4

Step 2: Perform the multiplication:
5×4=205 \times 4 = 20

Step 3: Perform the addition and subtraction from left to right:
63+20=236 - 3 + 20 = 23

The correct result is: 2323.

Answer

23 23

Exercise #8

7+495= 7 + \sqrt{49} - 5 =

Step-by-Step Solution

First, evaluate the square root: 49=7\sqrt{49}=7.

Then, follow the order of operations (PEMDAS/BODMAS):

1. Addition: 7+7=147 + 7 = 14

2. Subtraction: 145=914 - 5 = 9

So, the correct answer is 9 9 .

Answer

9 9

Exercise #9

8+3×242= 8 + 3 \times 2 - 4^2 =

Step-by-Step Solution

First, follow the order of operations (BODMAS/BIDMAS):

Step 1: Calculate the exponent:
42=164^2 = 16

Step 2: Perform the multiplication:
3×2=63 \times 2 = 6

Step 3: Perform the addition and subtraction from left to right:
8+616=1416=28 + 6 - 16 = 14 - 16 = -2

The correct result is: 2-2.

Answer

2 -2

Exercise #10

816×3= 8 - \sqrt{16} \times 3 =

Step-by-Step Solution

First, evaluate the square root: 16=4\sqrt{16}=4.

Then, follow the order of operations (PEMDAS/BODMAS):

1. Multiplication: 4×3=124 \times 3 = 12

2. Subtraction: 812=48 - 12 = -4

So, the correct answer is 4 -4 .

Answer

4 -4

Exercise #11

6+644= 6+\sqrt{64}-4=

Video Solution

Step-by-Step Solution

To solve the expression 6+644= 6+\sqrt{64}-4= , we need to follow the order of operations (PEMDAS/BODMAS):


  • P: Parentheses (or Brackets)
  • E: Exponents (or Orders, i.e., powers and roots, etc.)
  • MD: Multiplication and Division (left-to-right)
  • AS: Addition and Subtraction (left-to-right)

In this expression, we first need to evaluate the square root since it falls under the exponent category:


64=8 \sqrt{64} = 8


Next, we substitute the computed value back into the expression:


6+84 6+8-4


We then perform the addition and subtraction from left to right:


6+8=14 6+8 = 14


144=10 14-4 = 10


Thus, the final answer is:


10 10

Answer

10

Exercise #12

4+22= 4+2^2=

Video Solution

Step-by-Step Solution

To solve the expression 4+22 4 + 2^2 , follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).

Let's break down the expression:

  • Step 1: Identify any exponents.
    The expression contains an exponent: 22 2^2 . To evaluate this, multiply 2 by itself: 2×2 2 \times 2 , which equals 4.
    So, 22=4 2^2 = 4 .
  • Step 2: Perform addition.
    Now, substitute the result back into the original expression:
    4+4 4 + 4 .
    Add these numbers together: 4 + 4 equals 8.

Therefore, the answer to the expression 4+22 4 + 2^2 is 8.

Answer

8

Exercise #13

3×3+32= ? 3\times3+3^2=\text{ ?}

Video Solution

Step-by-Step Solution

First we need to remind ourselves of the order of operations:

  1. Parentheses

  2. Exponents and Roots

  3. Multiplication and Division

  4. Addition and Subtraction

There are no parentheses in this problem, therefore we will start with exponents:

3 * 3 + 3² =

3 * 3 + 9 =

Let's continue to the next step—multiplication operations:

3 * 3 + 9 =

9 + 9 =

Finally, we are left with a simple addition exercise:

9 + 9 = 18

Answer

18

Exercise #14

832:3= 8-3^2:3=

Video Solution

Step-by-Step Solution

Let's solve the expression step by step using the order of operations, often remembered by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

The given expression is: 832:3= 8-3^2:3=

Step 1: Evaluate Exponents
The expression has an exponent, which we need to evaluate first. The exponent is 323^2.
Calculate 323^2 which equals 99.
Now the expression becomes: 89:3 8 - 9 : 3

Step 2: Division
Next, perform the division operation. Here we divide 99 by 33.
Calculate 9:39 : 3 which equals 33.
Now the expression becomes: 83 8 - 3

Step 3: Subtraction
Finally, perform the subtraction.
Calculate 838 - 3 which equals 55.

Therefore, the solution to the expression 832:38-3^2:3 is 55.

Answer

5 5

Exercise #15

5+361= 5+\sqrt{36}-1=

Video Solution

Step-by-Step Solution

To solve the expression 5+361= 5+\sqrt{36}-1= , we need to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).


Here are the steps:


First, calculate the square root:

36=6 \sqrt{36} = 6

Substitute the square root back into the expression:

5+61 5 + 6 - 1

Next, perform the addition and subtraction from left to right:

Add 5 and 6:

5+6=11 5 + 6 = 11

Then subtract 1:

111=10 11 - 1 = 10

Finally, you obtain the solution:

10 10

Answer

10 10