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Order of Operations: Exponents and Radicals

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This lesson covers when and how to evaluate exponents and radicals using the order of operations.

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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:

PEMDAS: 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:

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:

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.

Exponents and radicals (roots, such as square roots and cube roots) are considered at the same time following the order of operations. 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.