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Order of Operations: Exponents and Radicals

Order of Operations: Exponents and Radicals

Author: Sophia Tutorial

Evaluate an expression with exponents or radicals using order of operations.

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what's covered
  1. Order of Operations: Exponents and Radicals

1. Order of Operations: Exponents and Radicals

The order of operations outlines an order we should follow to evaluate expressions with multiple operations. We often use PEMDAS to remember the order of operations:

term to know
An acronym to remember the order of operations: parentheses, exponents, multiplication and division, addition and subtraction.

The "exponents" part of PEMDAS also includes radicals. We don't typically include "radicals" in the acronym, because all radicals can be written as exponents (fractional exponents, specifically). So as long as we remember that exponents and radicals are related, we can remember to evaluate both exponents and radicals after parentheses, but before multiplication, division, addition, and subtraction.

Let's focus first on evaluating expressions containing exponents:

2 cubed plus 4 squared
8 plus 4 squared
Evaluate 2 cubed
8 plus 16
Evaluate 4 squared
Our Solution

25 minus 3 squared times 2
25 minus 9 times 2
Evaluate 3 squared first
25 minus 18
Then multiply 9 by 2
Our Solution

When we encounter radicals (roots, such as square roots or cube roots), we treat them as being on the same level as exponents in the order of operations, and evaluate them before multiplication, division, addition and subtraction. This is illustrated in the examples below:

15 plus square root of 16 times 2
15 plus 4 times 2
Evaluate the square root first
15 plus 8
Then multiply 4 by 2
Our Solution

6 times square root of 25 divided by 2
6 times 5 divided by 2
Evaluate the square root first
30 divided by 2
Then multiply 6 by 5
Our Solution

What do we do when an expression contains both exponents and radicals? Because exponents and radicals are at the same level according to the order of operations, we can think of their relationship being like multiplication and division, or addition and subtraction: we evaluate exponents and radicals as we see them reading left to right.

5 plus square root of 49 times 2 cubed
5 plus 7 times 2 cubed
Evaluate the square root first
5 plus 7 times 8
Then evaluate the exponent
5 plus 56
Multiply 7 by 8
Our Solution

When following order of operations, exponents and radicals (roots, such as square roots and cube roots) are considered at the same time. This means that we must evaluate all exponents and all radicals before moving on to the remaining operations in an expression, following the order of operations.