# 1.3 Radicals and rational exponents  (Page 4/11)

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## Understanding n Th roots

Suppose we know that $\text{\hspace{0.17em}}{a}^{3}=8.\text{\hspace{0.17em}}$ We want to find what number raised to the 3rd power is equal to 8. Since $\text{\hspace{0.17em}}{2}^{3}=8,$ we say that 2 is the cube root of 8.

The n th root of $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ is a number that, when raised to the n th power, gives $\text{\hspace{0.17em}}a.\text{\hspace{0.17em}}$ For example, $\text{\hspace{0.17em}}-3\text{\hspace{0.17em}}$ is the 5th root of $\text{\hspace{0.17em}}-243\text{\hspace{0.17em}}$ because $\text{\hspace{0.17em}}{\left(-3\right)}^{5}=-243.\text{\hspace{0.17em}}$ If $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ is a real number with at least one n th root, then the principal n th root    of $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ is the number with the same sign as $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ that, when raised to the n th power, equals $\text{\hspace{0.17em}}a.$

The principal n th root of $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ is written as $\text{\hspace{0.17em}}\sqrt[n]{a},$ where $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ is a positive integer greater than or equal to 2. In the radical expression, $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ is called the index    of the radical.

## Principal n Th root

If $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ is a real number with at least one n th root, then the principal n th root    of $\text{\hspace{0.17em}}a,$ written as $\text{\hspace{0.17em}}\sqrt[n]{a},$ is the number with the same sign as $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ that, when raised to the n th power, equals $\text{\hspace{0.17em}}a.\text{\hspace{0.17em}}$ The index    of the radical is $\text{\hspace{0.17em}}n.$

## Simplifying n Th roots

Simplify each of the following:

1. $\sqrt[5]{-32}$
2. $\sqrt[4]{4}\cdot \sqrt[4]{1,024}$
3. $-\sqrt[3]{\frac{8{x}^{6}}{125}}$
4. $8\sqrt[4]{3}-\sqrt[4]{48}$
1. $\sqrt[5]{-32}=-2\text{\hspace{0.17em}}$ because $\text{\hspace{0.17em}}{\left(-2\right)}^{5}=-32$
2. First, express the product as a single radical expression. $\text{\hspace{0.17em}}\sqrt[4]{4,096}=8\text{\hspace{0.17em}}$ because $\text{\hspace{0.17em}}{8}^{4}=4,096$

Simplify.

1. $\sqrt[3]{-216}$
2. $\frac{3\sqrt[4]{80}}{\sqrt[4]{5}}$
3. $6\sqrt[3]{9,000}+7\sqrt[3]{576}$
1. $-6$
2. $6$
3. $88\sqrt[3]{9}$

## Using 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 $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ is even, then $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ cannot be negative.

${a}^{\frac{1}{n}}=\sqrt[n]{a}$

We can also have rational exponents with numerators other than 1. In these cases, the exponent must be a fraction in lowest terms. We raise the base to a power and take an n th root. The numerator tells us the power and the denominator tells us the root.

${a}^{\frac{m}{n}}={\left(\sqrt[n]{a}\right)}^{m}=\sqrt[n]{{a}^{m}}$

All of the properties of exponents that we learned for integer exponents also hold for rational exponents.

## Rational exponents

Rational exponents are another way to express principal n th roots. The general form for converting between a radical expression with a radical symbol and one with a rational exponent is

${a}^{\frac{m}{n}}={\left(\sqrt[n]{a}\right)}^{m}=\sqrt[n]{{a}^{m}}$

Given an expression with a rational exponent, write the expression as a radical.

1. Determine the power by looking at the numerator of the exponent.
2. Determine the root by looking at the denominator of the exponent.
3. Using the base as the radicand, raise the radicand to the power and use the root as the index.

## Writing rational exponents as radicals

Write $\text{\hspace{0.17em}}{343}^{\frac{2}{3}}\text{\hspace{0.17em}}$ as a radical. Simplify.

The 2 tells us the power and the 3 tells us the root.

${343}^{\frac{2}{3}}={\left(\sqrt[3]{343}\right)}^{2}=\sqrt[3]{{343}^{2}}$

We know that $\text{\hspace{0.17em}}\sqrt[3]{343}=7\text{\hspace{0.17em}}$ because $\text{\hspace{0.17em}}{7}^{3}=343.\text{\hspace{0.17em}}$ Because the cube root is easy to find, it is easiest to find the cube root before squaring for this problem. In general, it is easier to find the root first and then raise it to a power.

${343}^{\frac{2}{3}}={\left(\sqrt[3]{343}\right)}^{2}={7}^{2}=49$

Write $\text{\hspace{0.17em}}{9}^{\frac{5}{2}}\text{\hspace{0.17em}}$ as a radical. Simplify.

${\left(\sqrt{9}\right)}^{5}={3}^{5}=243$

## Writing radicals as rational exponents

Write $\text{\hspace{0.17em}}\frac{4}{\sqrt[7]{{a}^{2}}}\text{\hspace{0.17em}}$ using a rational exponent.

The power is 2 and the root is 7, so the rational exponent will be $\text{\hspace{0.17em}}\frac{2}{7}.\text{\hspace{0.17em}}$ We get $\text{\hspace{0.17em}}\frac{4}{{a}^{\frac{2}{7}}}.\text{\hspace{0.17em}}$ Using properties of exponents, we get $\text{\hspace{0.17em}}\frac{4}{\sqrt[7]{{a}^{2}}}=4{a}^{\frac{-2}{7}}.$

Write $\text{\hspace{0.17em}}x\sqrt{{\left(5y\right)}^{9}}\text{\hspace{0.17em}}$ using a rational exponent.

$x{\left(5y\right)}^{\frac{9}{2}}$

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
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