Rational exponents
If $b$ is an $n$th root of $a$, then,
$b=\sqrt[n]{a}$
and
$a=b^n$
Now, if we raise both sides to the power $1/n$, we get
$\begin{align*} a^{1/n}&=(b^n)^{1/n}\\ &=b\\ &=\sqrt[n]{a}\\ \end{align*}$
That is, $\boxed{\sqrt[n]{a}=a^{1/n}}$
The left-hand side is in radical form, and the right-hand side is in rational-exponent form. We can switch between exponential and radical form using this relation.
Further, we can write
$\begin{align*} a^{m/n}&=(a^m)^{1/n}\\ &=\sqrt[n]{a^m}\\ \end{align*}$
and
$\begin{align*} a^{m/n}&=(a^{1/n})^m\\ &=(\sqrt[n]{a})^m\\ \end{align*}$
Combining the above two relations, we can write
$a^{m/n}=\sqrt[n]{a^m}=(\sqrt[n]{a})^m$
Example 1: Write the following radical expressions as exponential expressions: (a) $\sqrt{13}$; (b) $\sqrt[5]{7}$
Solution:
(a). $\sqrt{13}=13^{1/2}$
(b). $\sqrt[5]{7}=7^{1/5}$
Example 2: Write the radical expression $\sqrt[7]{3^4}$ as an exponential expression.
Solution:
$\begin{align*} \sqrt[7]{3^4}&=(3^4)^{1/7}\\ &=3^{4\cdot 1/7}\\ &=3^{4/7}\\ \end{align*}$
Simplifying rational exponents
Example 1: Simplify (write your answer without exponents), $\left(\dfrac{1}{9}\right)^{3/2}$.
Solution:
$\begin{align*} \left(\dfrac{1}{9}\right)^{3/2}&=\left(\dfrac{1}{9}\right)^{3/2}\\ &=\dfrac{1^{3/2}}{9^{3/2}}\\ &=\dfrac{1}{(3^2)^{3/2}}\\ &=\dfrac{1}{27}\\ \end{align*}$
Example 2: Simplify (write your answer without exponents), $32^{-\tfrac{4}{5}}$.
Solution:
First, make the exponent positive by taking the reciprocal of the base.
$\begin{align*} 32^{-\tfrac{4}{5}} &= \left(\dfrac{1}{32}\right)^{\tfrac{4}{5}}\\ &=\dfrac{1}{32^{\tfrac{4}{5}}}\\ &=\dfrac{1}{(2^5)^{\tfrac{4}{5}}}\\ &=\dfrac{1}{2^4}\\ &=\dfrac{1}{16} \end{align*}$
Rational exponents: product rule
You have already learned that if $m$ and $n$ are integers, then
$a^m\cdot a^n=a^{m+n}$
The same rule applies when $m$ and $n$ are fractions rather than integers.
Rational exponents: quotient rule
You have already learned that if $m$ and $n$ are integers, then
$\dfrac{a^m}{a^n}=a^{m-n}$
The same rule applies when $m$ and $n$ are fractions rather than integers.
Example 1: Simplify $x^{7/3}\cdot x^{2/5}$. Write your answer with positive exponents.
Solution:
$\begin{align*} x^{7/3}\cdot x^{2/5}&=x^{7/3+2/5}\\ &=x^{41/15}\\ \end{align*}$
Example 2: Simplify $\dfrac{u^{11/4}}{u^{3/5}}.$ Write your answer with positive exponents.
Solution:
$\begin{align*} \dfrac{u^{11/4}}{u^{3/5}}&=u^{11/4}-u^{3/5}\\ &=u^{11/4-3/5}\\ &=u^{43/20}\\ \end{align*}$
Example 3: Simplify $\dfrac{x^{-4/3}\:x^{2/5}}{x^{7/6}}$. Write your answer with positive exponents.
Solution:
$ \begin{align*} \dfrac{x^{-4/3}\:x^{2/5}}{x^{7/6}} &=\dfrac{x^{-4/3+2/5}}{x^{7/6}}\\ &=\dfrac{x^{-14/15}}{x^{7/6}}\\ &=\dfrac{1}{x^{7/6+14/15}}\\ &=\dfrac{1}{x^{21/10}}\\ \end{align*}$
Rational exponents: power of a power rule
If $m$ and $n$ are positive integers, then you know that
$(a^m)^n=a^{mn}$
This formula is valid not only for integer $m$ and $n$, but for fractions too.
Example 1: Simplify $\bigg(x^\tfrac{5}{8}\bigg)^2$. Assume $x$ represents a positive real number.
Solution:
$\begin{align*} \left (x^\tfrac{5}{8}\right )^2 &= \left(x^\tfrac{5}{8}\right)^2\\ &= x^{\tfrac{5}{8}\cdot 2}\\ &= x^\tfrac{5}{4}\\ \end{align*}$
Example 2: Simplify $\bigg(y^\tfrac{4}{3}\bigg)^{\tfrac{3}{2}}$. Assume $y$ represents a positive real number.
Solution:
$\begin{align*} \bigg(y^\tfrac{4}{3}\bigg)^{\tfrac{3}{2}} &= y^{\tfrac{4}{3}\cdot \tfrac{3}{2}}\\ &= y^2\\ \end{align*}$
Rational exponents: power of product and power of quotient rules
$(ab)^m=a^m\:b^m$
$\left(\dfrac{a}{b}\right)^m=\dfrac{a^m}{b^m}$
Example: Simplify $\left(x^{\tfrac{4}{7}}y^{\tfrac{2}{5}}\right)^{-\tfrac{2}{3}}$. Assume the variables represent positive real numbers.
Solution:
$\begin{align*} \left(x^{\tfrac{4}{7}}y^{\tfrac{2}{5}}\right)^{-\tfrac{2}{3}} &= \dfrac{1}{\left(x^{\tfrac{4}{7}}y^{\tfrac{2}{5}}\right)^{\tfrac{2}{3}}} \\ &=\dfrac{1}{x^{\tfrac{4}{7}\cdot \tfrac{2}{3}}y^{\tfrac{2}{5}\cdot \tfrac{2}{3}}} \\ &=\dfrac{1}{x^{\tfrac{8}{21}}y^{\tfrac{4}{15}}} \\ \end{align*}$
Multiplication property of radicals
$\sqrt[n]{ab}=\sqrt[n]{a}\:\sqrt[n]{b}$
In the following examples, you will see how to simplify radicals using the multiplication property.
Example 1: Simplify $\sqrt{50y^7z^3}$
Solution:
$\begin{align*} \sqrt{50y^7z^3}&= \sqrt{2\cdot 25y^6yz^2z}\\ &=\sqrt{25}\:\sqrt{y^6}\:\sqrt{z^2}\sqrt{2yz}\\ &=5y^3z\sqrt{2yz}\\ \end{align*}$
Example 2: Simplify $\sqrt[3]{192x^{10}y^3}$
Solution:
$\begin{align*} \sqrt[3]{192x^{10}y^3}&=\sqrt[3]{192x^{10}y^3}\\ &=\sqrt[3]{3\cdot 64 x^9 x y^3}\\ &=\sqrt[3]{64}\:\sqrt[3]{x^9}\:\sqrt[3]{y^3}\:\sqrt[3]{3x}\\ &=4x^3y\:\sqrt[3]{3x}\\ \end{align*}$
Example 3: Simplify $\sqrt[4]{112u^{15}w^9}$
Solution:
$\begin{align*} \sqrt[4]{112u^{15}w^9} &= \sqrt[4]{7\cdot 16 u^{12}u^3w^8w}\\ &= \sqrt[4]{16}\:\sqrt[4]{u^{12}}\:\sqrt[4]{w^8}\sqrt[4]{7u^3w}\\ &= 2u^3w^2\:\sqrt[4]{7u^3w}\\ \end{align*}$
Addition and subtraction of radicals
We can combine like radicals just as we combine like terms. Like radicals are radicals with the same index and the same radicand. The radicand is the expression under the radical sign.
Examples:
$4\sqrt{11}+3\sqrt{11}=7\sqrt{11}$
$2x\sqrt[5]{yz}+3x\sqrt[5]{yz}=5x\sqrt[5]{yz}$
Example 1: Simplify, $3\sqrt{20}+6\sqrt{45}-8\sqrt{5}$
Solution:
$\begin{align*} 3\sqrt{20}+6\sqrt{45}-8\sqrt{5}&=3\sqrt{4\cdot 5}+6\sqrt{9\cdot 5}-8\sqrt{5}\\ &=3\cdot 2\sqrt{5}+6\cdot 3 \sqrt{5}-8\sqrt{5}\\ &=6\sqrt{5}+18 \sqrt{5}-8\sqrt{5}\\ &=16\sqrt{5}\\ \end{align*}$
Example 2: Simplify $x^3\sqrt{75y^3}-2y\sqrt{108x^6y}$
Solution:
$\begin{align*} x^3\sqrt{75y^3}-2y\sqrt{108x^6y}&=x^3\sqrt{3\cdot 25 y^2y}-2y\sqrt{3\cdot 36 x^6y}\\ &=x^3\cdot 5y\sqrt{3y}-2y\cdot 6x^3\sqrt{3y}\\ &=5x^3y\sqrt{3y}-12x^3y\sqrt{3y}\\ &=-7x^3y\sqrt{3y}\\ \end{align*}$
Example 3: Simplify $2y\sqrt[3]{54u^3y^8}+8u\sqrt[3]{250y^{11}}$
Solution:
$\begin{align*} 2y\sqrt[3]{54u^3y^8}+8u\sqrt[3]{250y^{11}}&=2y\sqrt[3]{2\cdot 27u^3y^6y^2}+8u\sqrt[3]{2\cdot 125y^{9}y^2}\\ &=2y\cdot 3uy^2\sqrt[3]{2y^2}+8u\cdot 5y^3\sqrt[3]{2y^2}\\ &=6uy^3\sqrt[3]{2y^2}+40uy^3\sqrt[3]{2y^2}\\ &=46uy^3\sqrt[3]{2y^2}\\ \end{align*}$