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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, $\text{\hspace{0.17em}}\mathrm{-3}\text{\hspace{0.17em}}$ is the 5th root of $\text{\hspace{0.17em}}\mathrm{-243}\text{\hspace{0.17em}}$ because $\text{\hspace{0.17em}}{(\mathrm{-3})}^{5}=\mathrm{-243.}\text{\hspace{0.17em}}$ 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.
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 $\text{\hspace{0.17em}}n.$
Simplify each of the following:
Simplify.
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.
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.
All of the properties of exponents that we learned for integer exponents also hold for 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
Given an expression with a rational exponent, write the expression as a radical.
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.
${\left(\sqrt{9}\right)}^{5}={3}^{5}=243$
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 $\text{\hspace{0.17em}}\frac{4}{\sqrt[7]{{a}^{2}}}=4{a}^{\frac{\mathrm{-2}}{7}}.$
Write $\text{\hspace{0.17em}}x\sqrt{{(5y)}^{9}}\text{\hspace{0.17em}}$ using a rational exponent.
$x{(5y)}^{\frac{9}{2}}$
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