Understanding nth Roots

Suppose we know that [latex]{a}^{3}=8[/latex]. We want to find what number raised to the 3rd power is equal to 8. Since [latex]{2}^{3}=8[/latex], we say that 2 is the cube root of 8.

The nth root of [latex]a[/latex] is a number that, when raised to the nth power, gives [latex]a[/latex]. For example, [latex]-3[/latex] is the 5th root of [latex]-243[/latex] because [latex]{\left(-3\right)}^{5}=-243[/latex]. If [latex]a[/latex] is a real number with at least one nth root, then the principal nth root of [latex]a[/latex] is the number with the same sign as [latex]a[/latex] that, when raised to the nth power, equals [latex]a[/latex].

The principal nth root of [latex]a[/latex] is written as [latex]\sqrt[n]{a}[/latex], where [latex]n[/latex] is a positive integer greater than or equal to 2. In the radical expression, [latex]n[/latex] is called the index of the radical.

A General Note: Principal nth Root

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

Example 10: Simplifying nth Roots

Simplify each of the following:

  1. [latex]\sqrt[5]{-32}[/latex]
  2. [latex]\sqrt[4]{4}\cdot \sqrt[4]{1,024}[/latex]
  3. [latex]-\sqrt[3]{\frac{8{x}^{6}}{125}}[/latex]
  4. [latex]8\sqrt[4]{3}-\sqrt[4]{48}[/latex]

Solution

  1. [latex]\sqrt[5]{-32}=-2[/latex] because [latex]{\left(-2\right)}^{5}=-32 \\ \text{ }[/latex]
  2. First, express the product as a single radical expression. [latex]\sqrt[4]{4,096}=8[/latex] because [latex]{8}^{4}=4,096 \\[/latex]
  3. [latex]\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}[/latex]
  4. [latex]\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}[/latex]

Try It 10

Simplify.

  1. [latex]\sqrt[3]{-216}[/latex]
  2. [latex]\frac{3\sqrt[4]{80}}{\sqrt[4]{5}}[/latex]
  3. [latex]6\sqrt[3]{9,000}+7\sqrt[3]{576}[/latex]

Solution