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Study Guides > College Algebra

Understanding nth Roots

Suppose we know that a3=8{a}^{3}=8. We want to find what number raised to the 3rd power is equal to 8. Since 23=8{2}^{3}=8, we say that 2 is the cube root of 8. The nth root of aa is a number that, when raised to the nth power, gives aa. For example, 3-3 is the 5th root of 243-243 because (3)5=243{\left(-3\right)}^{5}=-243. If aa is a real number with at least one nth root, then the principal nth root of aa is the number with the same sign as aa that, when raised to the nth power, equals aa. The principal nth root of aa is written as an\sqrt[n]{a}, where nn is a positive integer greater than or equal to 2. In the radical expression, nn is called the index of the radical.

A General Note: Principal nth Root

If aa is a real number with at least one nth root, then the principal nth root of aa, written as an\sqrt[n]{a}, is the number with the same sign as aa that, when raised to the nth power, equals aa. The index of the radical is nn.

Example 10: Simplifying nth Roots

Simplify each of the following:
  1. 325\sqrt[5]{-32}
  2. 441,0244\sqrt[4]{4}\cdot \sqrt[4]{1,024}
  3. 8x61253-\sqrt[3]{\frac{8{x}^{6}}{125}}
  4. 8344848\sqrt[4]{3}-\sqrt[4]{48}

Solution

  1. 325=2\sqrt[5]{-32}=-2 because (2)5=32 {\left(-2\right)}^{5}=-32 \\ \text{ }
  2. First, express the product as a single radical expression. 4,0964=8\sqrt[4]{4,096}=8 because 84=4,096{8}^{4}=4,096 \\
  3. 8x631253Write as quotient of two radical expressions.2x25Simplify.\begin{array}{cc}\\ \frac{-\sqrt[3]{8{x}^{6}}}{\sqrt[3]{125}}\hfill & \text{Write as quotient of two radical expressions}.\hfill \\ \frac{-2{x}^{2}}{5}\hfill & \text{Simplify}.\hfill \\ \end{array}
  4. 834234Simplify to get equal radicands.634Add.\begin{array}{cc}\\ 8\sqrt[4]{3}-2\sqrt[4]{3}\hfill & \text{Simplify to get equal radicands}.\hfill \\ 6\sqrt[4]{3} \hfill & \text{Add}.\hfill \\ \end{array}

Try It 10

Simplify.
  1. 2163\sqrt[3]{-216}
  2. 380454\frac{3\sqrt[4]{80}}{\sqrt[4]{5}}
  3. 69,0003+757636\sqrt[3]{9,000}+7\sqrt[3]{576}
Solution

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  • College Algebra. Provided by: OpenStax Authored by: OpenStax College Algebra. Located at: https://cnx.org/contents/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1/Preface. License: CC BY: Attribution.