Although square roots are the most commonly recognized, we can also find cube roots, fourth roots, fifth roots, and beyond. Just as the square root function reverses the squaring function, these higher-order roots are the inverses of their corresponding power functions. These root functions are particularly useful when you need to identify the number that, when raised to a specific power, results in a given number.
[latex]\text{n}^{\text{th}}[/latex] Root
The [latex]\text{n}^{\text{th}}[/latex] root of [latex]a[/latex], denoted as [latex]\sqrt[n]{a}[/latex], is a number that, when raised to the nth power, equals [latex]a[/latex].
For example, if you are looking for the cube root ([latex]3^{\text{rd}}[/latex] root) of [latex]8[/latex], you need to find a number that, when multiplied by itself three times, gives [latex]8[/latex]. In this case, [latex]2 \cdot 2 \cdot 2 = 8[/latex], so the cube root of [latex]8[/latex] is [latex]2[/latex]. That is, [latex]\sqrt[3]{8} = 2[/latex].
Now that we’ve explored [latex]\text{n}^{\text{th}}[/latex] roots, let’s look into a specific type of [latex]\text{n}^{\text{th}}[/latex] root known as the principal [latex]\text{n}^{\text{th}}[/latex] root. When we talk about the principal [latex]\text{n}^{\text{th}}[/latex] root, we’re focusing on the most common or main root that we use in math, especially when the root needs to be positive.
principal [latex]\text{n}^{\text{th}}[/latex] root
If [latex]a[/latex] is a real number with at least one [latex]\text{n}^{\text{th}}[/latex] root, then the principal [latex]\text{n}^{\text{th}}[/latex] 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 [latex]\text{n}^{\text{th}}[/latex] power, equals [latex]a[/latex].
Here, [latex]n[/latex] is what we call the index of the radical, which tells us the degree of the root.Simplify each of the following:
[latex]\sqrt[5]{-32}[/latex]
[latex]\sqrt[4]{4}\cdot \sqrt[4]{1,024}[/latex]
[latex]-\sqrt[3]{\dfrac{8{x}^{6}}{125}}[/latex]
[latex]8\sqrt[4]{3}-\sqrt[4]{48}[/latex]
[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\text{,}096}=8[/latex] because [latex]{8}^{4}=4,096[/latex]
[latex]\begin{align} -\sqrt[3]{\frac{8x^6}{125}} &= \frac{-\sqrt[3]{8x^6}}{\sqrt[3]{125}} && \text{Write as quotient of two radical expressions.} \\ &= \frac{-2x^2}{5} && \text{Simplify.} \end{align}[/latex]
[latex]\begin{align}\\ 8\sqrt[4]{3}-\sqrt[4]{48} & = 8\sqrt[4]{3}-2\sqrt[4]{3} && \text{Simplify to get equal radicands}. \\ & = 6\sqrt[4]{3} && \text{Add}. \end{align}[/latex]
Rational Exponents
Radical expressions can also be written without using the radical symbol. We can use rational (fractional) exponents. The index must be a positive integer. If the index [latex]n[/latex] is even, then [latex]x[/latex] cannot be negative.
Radical Form
Exponent Form
[latex]\sqrt{x}[/latex]
[latex]x^{\frac{1}{2}}[/latex]
[latex]\sqrt[3]{x}[/latex]
[latex]x^{\frac{1}{3}}[/latex]
[latex]\sqrt[4]{x}[/latex]
[latex]x^{\frac{1}{4}}[/latex]
…
…
[latex]\sqrt[n]{x}[/latex]
[latex]x^{\frac{1}{n}}[/latex]
We can also have rational exponents with numerators other than 1.
rational exponents
Rational exponents are another way to express principal [latex]\text{n}^{\text{th}}[/latex] roots.
The general form for converting between a radical expression with a radical symbol and one with a rational exponent is
