# 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*n*

^{th}root of [latex]a[/latex] is a number that, when raised to the

*n*

^{th}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

*n*

^{th}root, then the

**principal**of [latex]a[/latex] is the number with the same sign as [latex]a[/latex] that, when raised to the

*n*^{th}root*n*

^{th}power, equals [latex]a[/latex]. The principal

*n*

^{th}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 *n*th Root

If [latex]a[/latex] is a real number with at least one *n*

^{th}root, then the

**principal**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

*n*^{th}root*n*

^{th}power, equals [latex]a[/latex]. The

**index**of the radical is [latex]n[/latex].

### Example 10: Simplifying *n*th Roots

Simplify each of the following:
- [latex]\sqrt[5]{-32}[/latex]
- [latex]\sqrt[4]{4}\cdot \sqrt[4]{1,024}[/latex]
- [latex]-\sqrt[3]{\frac{8{x}^{6}}{125}}[/latex]
- [latex]8\sqrt[4]{3}-\sqrt[4]{48}[/latex]

### Solution

- [latex]\sqrt[5]{-32}=-2[/latex] because [latex]{\left(-2\right)}^{5}=-32 \\ \text{ }[/latex]
- First, express the product as a single radical expression. [latex]\sqrt[4]{4,096}=8[/latex] because [latex]{8}^{4}=4,096 \\[/latex]
- [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]
- [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.- [latex]\sqrt[3]{-216}[/latex]
- [latex]\frac{3\sqrt[4]{80}}{\sqrt[4]{5}}[/latex]
- [latex]6\sqrt[3]{9,000}+7\sqrt[3]{576}[/latex]

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