Let's start with something "easy":
The square root of $x$ is a number such that when we square it we get $x$.
Example: $\sqrt{9}$ is a positive number such that when we square it we get $9.$
In general, $y=\sqrt{x}$ is (almost) the same as saying $y^2=x.$
A Little Bit Harder
The cube root of $x$ is a number such that when we cube it we get $x$.
Example $\sqrt[3]{-8}$ is a number such that when we cube it we get $-8$.
In general, $y=\sqrt[3]{x}$ really is the same as saying $y^3=x.$
The Big Picture: General Radicals
The $n$th root of $x$ is a number such that when we raise it to the $n$th power we get $x$.
In general, $y=\sqrt[n]{x}$ is (ALMOST) the same as $y^n=x.$
Vocab: $n$ is called the index of the radical.
Vocab: The quantity under the radical $x$ is called the radicand.
Example : $\sqrt[6]{64}$ is a number such that when we raise it to the $6$th power we get $64$.
Here the index is $6$ and the radicand is $64.$
Examples: Evaluate the radical expressions.
$\sqrt{36}$
$\sqrt[3]{27}$
$\sqrt[3]{64}+\sqrt[3]{-1}$
$\sqrt[4]{81}$
$\sqrt[5]{32}$
Weird Examples: Evaluate the radical expressions.
$\sqrt{-9}$
$-\sqrt{9}$
$\sqrt[3]{0}$
$\sqrt[4]{1}$
$\sqrt[5]{-1}$
Examples: Simplifying Variable Expressions with Roots
Method 1: The Power Method.
$\sqrt{64x^2}$
$$
\begin{array}{lll}
\displaystyle \sqrt{64x^2}&\displaystyle=\sqrt{(8x)^2} &\mbox{}\\
\displaystyle &\displaystyle=8x &\mbox{}\\
\end{array}
$$
$\sqrt[3]{8 x^{30}y^{12}}$
$$
\begin{array}{lll}
\displaystyle \sqrt[3]{8 x^{30}y^{12}}&\displaystyle=\sqrt[3]{\left(8 x^{10}y^{4}\right)^3} &\mbox{}\\
\displaystyle &\displaystyle=8 x^{10}y^{4} &\mbox{}\\
\end{array}
$$
$-\sqrt[4]{81p^{28}q^{16}}$
$$
\begin{array}{lll}
\displaystyle -\sqrt[4]{81p^{28}q^{16}}&\displaystyle=-\sqrt[4]{\left(3p^{7}q^{4}\right)^4} &\mbox{}\\
\displaystyle &\displaystyle=-3p^{7}q^{4} &\mbox{}\\
\end{array}
$$
Big Fact About Radicals $$\sqrt[n]{ab}=\sqrt[n]{a}\sqrt[n]{b}$$ Example: Use the big fact to simplify $\sqrt{75}.$
Method 1: Power Method
$$ $$ \begin{array}{lll} \displaystyle \sqrt{75}&\displaystyle=\sqrt{25\cdot 3} &\mbox{factor out perfect squares}\\ \displaystyle &\displaystyle=\sqrt{25}\sqrt{3} &\mbox{use Big Fact: a radical of the product is a product of the radicals}\\ \displaystyle &\displaystyle=5\sqrt{3} &\mbox{simplify}\\ \end{array} $$ $$ Method 2: Jail-Break Method
$$ \begin{array}{lll} \displaystyle \sqrt{75}&\displaystyle=\sqrt{5 \cdot 5 \cdot 3} &\mbox{factor completely}\\ \displaystyle &\displaystyle=\sqrt{\underline{5 \cdot 5} \cdot 3} &\mbox{underline pairs}\\ \displaystyle &\displaystyle=5\sqrt{3} &\mbox{pairs break out of square root jail}\\ \end{array} $$
$$ $$ \begin{array}{lll} \displaystyle \sqrt{75}&\displaystyle=\sqrt{25\cdot 3} &\mbox{factor out perfect squares}\\ \displaystyle &\displaystyle=\sqrt{25}\sqrt{3} &\mbox{use Big Fact: a radical of the product is a product of the radicals}\\ \displaystyle &\displaystyle=5\sqrt{3} &\mbox{simplify}\\ \end{array} $$ $$ Method 2: Jail-Break Method
$$ \begin{array}{lll} \displaystyle \sqrt{75}&\displaystyle=\sqrt{5 \cdot 5 \cdot 3} &\mbox{factor completely}\\ \displaystyle &\displaystyle=\sqrt{\underline{5 \cdot 5} \cdot 3} &\mbox{underline pairs}\\ \displaystyle &\displaystyle=5\sqrt{3} &\mbox{pairs break out of square root jail}\\ \end{array} $$
Good News!
As the above example suggests, we can use The Power Method or The Jail-Break Method in conjunction with The Big Fact to simplify more general kinds of radical expressions.
Example: Use the Power Method and the Big Fact to simplify $\sqrt{55k^{5}}.$
$$
\begin{array}{lll}
\displaystyle \sqrt{55k^{5}}&\displaystyle=\sqrt{k^4\cdot 55k} &\mbox{}\\
\displaystyle &\displaystyle=\sqrt{k^4}\sqrt{55k} &\mbox{a radical of a product is the product of the radicals!}\\
\displaystyle &\displaystyle=\sqrt{\left(k^2\right)^2}\sqrt{55k} &\mbox{}\\
\displaystyle &\displaystyle=k^2\sqrt{55k}&\mbox{}\\
\end{array}
$$
Example: Use the Jail-Break Method and the Big Fact to simplify $\sqrt[3]{432x^{3}p^{2}}$
$$
\begin{array}{lll}
\displaystyle \sqrt[3]{432x^{3}p^{2}}&\displaystyle=\sqrt[3]{2\cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot x \cdot x \cdot x \cdot p \cdot p} &\mbox{factor completely}\\
\displaystyle &\displaystyle=\sqrt[3]{\underline{2\cdot 2 \cdot 2} \cdot 2 \cdot \underline{3 \cdot 3 \cdot 3} \cdot \underline{x \cdot x \cdot x} \cdot p \cdot p} &\mbox{underline triplets}\\
\displaystyle &\displaystyle=2\cdot 3 \cdot x \cdot \sqrt[3]{2 \cdot p \cdot p} &\mbox{each triplet sends out a representative}\\
\displaystyle &\displaystyle=6x\sqrt[3]{2p^2} &\mbox{simplify}\\
\end{array}
$$
More Examples!
$\sqrt[3]{56x^5y^4}$
$$
\begin{array}{lll}
\displaystyle \sqrt[3]{56x^5y^4}&\displaystyle=\sqrt[3]{2 \cdot 2 \cdot 2 \cdot 7 \cdot x \cdot x \cdot x \cdot x \cdot x \cdot y \cdot y \cdot y \cdot y} &\mbox{factor completely}\\
\displaystyle &\displaystyle=\sqrt[3]{\underline{2 \cdot 2 \cdot 2} \cdot 7 \cdot \underline{x \cdot x \cdot x} \cdot x \cdot x \cdot \underline{y \cdot y \cdot y} \cdot y} &\mbox{underline triplets}\\
\displaystyle &\displaystyle= 2 \cdot x \cdot y\sqrt[3]{7\cdot x \cdot x \cdot y}&\mbox{each triplet sends out a representative}\\
\displaystyle &\displaystyle= 2xy\sqrt[3]{7x^2y}&\mbox{simplify}\\
\end{array}
$$
$\sqrt[4]{4320}$
$$
\begin{array}{lll}
\displaystyle \sqrt[4]{4320}&\displaystyle=\sqrt[4]{2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3\cdot 3\cdot 5} &\mbox{factor completely}\\
\displaystyle &\displaystyle=\sqrt[4]{\underline{2 \cdot 2 \cdot 2 \cdot 2} \cdot 2 \cdot 3 \cdot 3\cdot 3\cdot 5} &\mbox{underline quadruplets}\\
\displaystyle &\displaystyle=2\sqrt[4]{ 2 \cdot 3 \cdot 3\cdot 3\cdot 5} &\mbox{each quadruplet sends out a representative}\\
\displaystyle &\displaystyle=2\sqrt[4]{270} &\mbox{simplify}\\
\end{array}
$$
$\sqrt[4]{80a^7b^4}$
$$
\begin{array}{lll}
\displaystyle \sqrt[4]{80a^7b^4}&\displaystyle=\sqrt[4]{\left(16 a^4b^4\right)\left(2a^3\right)} &\mbox{}\\
\displaystyle &\displaystyle=\sqrt[4]{\left(2ab\right)^4\left(2a^3\right)} &\mbox{}\\
\displaystyle &\displaystyle=\sqrt[4]{\left(2ab\right)^4}\sqrt[4]{2a^3} &\mbox{a radical of a product is the product of the radicals!}\\
\displaystyle &\displaystyle=2ab\sqrt[4]{2a^3} &\mbox{}\\
\end{array}
$$
Another Big Fact About Radicals $$\sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}$$ Example: Use this big fact to simplify $\displaystyle \sqrt[3]{\frac{8}{125}}.$
$$
\begin{array}{lll}
\displaystyle \sqrt[3]{\frac{8}{125}}&\displaystyle=\frac{\sqrt[3]{8}}{\sqrt[3]{125}} &\mbox{a radical of a quotient is the quotient of the radicals!}\\
\displaystyle &\displaystyle=\frac{2}{5} &\mbox{}\\
\end{array}
$$
Examples: Radicals of Quotients
$\displaystyle \sqrt{\frac{98}{162}}$
$$
\begin{array}{lll}
\displaystyle \sqrt{\frac{98}{162}}&\displaystyle=\sqrt{\frac{49}{81}} &\mbox{simplfy fraction first}\\
\displaystyle &\displaystyle=\frac{\sqrt{49}}{\sqrt{81}} &\mbox{a radical of a quotient is the quotient of the radicals!}\\
\displaystyle &\displaystyle=\frac{7}{9} &\mbox{}\\
\end{array}
$$
$\displaystyle \sqrt[4]{\frac{162m^{14}}{n^{12}}}$
$$
\begin{array}{lll}
\displaystyle \sqrt[4]{\frac{162m^{14}}{n^{12}}}&\displaystyle=\frac{\sqrt[4]{162m^{14}}}{\sqrt[4]{n^{12}}} &\mbox{a radical of a quotient is the quotient of the radicals!}\\
\displaystyle &\displaystyle=\frac{\sqrt[4]{2\cdot 3^4 m^{12}m^2}}{n^3} &\mbox{}\\
\displaystyle &\displaystyle=\frac{\sqrt[4]{(3^4 m^{12})(2m^2) }}{n^3} &\mbox{}\\
\displaystyle &\displaystyle=\frac{\sqrt[4]{3^4 m^{12}}\sqrt[4]{2m^2}}{n^3} &\mbox{a radical of a product is the product of the radicals!}\\
\displaystyle &\displaystyle=\frac{3m^3\sqrt[4]{2m^2}}{n^3} &\mbox{}\\
\end{array}
$$
$\displaystyle \sqrt[3]{\frac{54x^2y^5}{250x^7y^2}}$
$$
\begin{array}{lll}
\displaystyle \sqrt[3]{\frac{54x^2y^5}{250x^7y^2}}&\displaystyle= \sqrt[3]{\frac{27y^3}{125x^5}}&\mbox{simplify fraction}\\
\displaystyle &\displaystyle= \frac{\sqrt[3]{27y^3}}{\sqrt[3]{125x^5}}&\mbox{a radical of a quotient is the quotient of the radicals!}\\
\displaystyle &\displaystyle= \frac{\sqrt[3]{27}\sqrt[3]{y^3}}{\sqrt[3]{125}\sqrt[3]{x^5}}&\mbox{a radical of a product is the product of the radicals!}\\
\displaystyle &\displaystyle= \frac{3y}{5x^{\color{magenta}{1}}\sqrt[3]{x^{\color{blue}{2}}}}&\mbox{3 goes into 5 with quotient $\color{magenta}{1}$ and remainder $\color{blue}{2}$}\\
\displaystyle &\displaystyle= \frac{3y}{5x\sqrt[3]{x^2}}&\mbox{}\\
\end{array}
$$
More Examples: Quotients of Radicals
$\displaystyle \frac{\sqrt{128z^9}}{\sqrt{2z}}$ (Assume all variables are nonnegative.)
$$
\begin{array}{lll}
\displaystyle \frac{\sqrt{128z^9}}{\sqrt{2z}}&\displaystyle=\sqrt{\frac{128z^9}{2z}} &\mbox{a quotient of radicals is a radical of the quotient!}\\
\displaystyle &\displaystyle=\sqrt{\frac{64z^8}{1}} &\mbox{}\\
\displaystyle &\displaystyle=\sqrt{64z^8} &\mbox{}\\
\displaystyle &\displaystyle=\sqrt{64}\sqrt{z^8} &\mbox{a radical of the product is the product of the radicals!}\\
\displaystyle &\displaystyle=8z^4 &\mbox{}\\
\end{array}
$$
$\displaystyle \frac{\sqrt[3]{-500}}{\sqrt[3]{2}}$
$$
\begin{array}{lll}
\displaystyle \frac{\sqrt[3]{-500}}{\sqrt[3]{2}}&\displaystyle=\sqrt[3]{\frac{-500}{2}} &\mbox{a quotient of radicals is a radical of the quotient!}\\
\displaystyle &\displaystyle=\sqrt[3]{-250} &\mbox{}\\
\displaystyle &\displaystyle=\sqrt[3]{-125\cdot 2} &\mbox{}\\
\displaystyle &\displaystyle=\sqrt[3]{-125}\sqrt[3]{2} &\mbox{a radical of the product is the product of the radicals!}\\
\displaystyle &\displaystyle=-5\sqrt[3]{2} &\mbox{}\\
\end{array}
$$
$\displaystyle \frac{\sqrt[4]{486m^{11}}}{\sqrt[4]{3m^5}}$ (Assume all variables are nonnegative.)
$$
\begin{array}{lll}
\displaystyle \frac{\sqrt[4]{486m^{11}}}{\sqrt[4]{3m^5}}&\displaystyle=\sqrt[4]{\frac{486m^{11}}{3m^5}} &\mbox{a quotient of radicals is a radical of the quotient!}\\
\displaystyle &\displaystyle=\sqrt[4]{162m^{6}} &\mbox{}\\
\displaystyle &\displaystyle=\sqrt[4]{2\cdot 81m^{4}m^2} &\mbox{}\\
\displaystyle &\displaystyle=\sqrt[4]{81m^{4}\cdot 2m^2 } &\mbox{}\\
\displaystyle &\displaystyle=\sqrt[4]{81}\sqrt[4]{m^{4}} \sqrt[4]{2m^2 } &\mbox{a radical of the product is the product of the radicals!}\\
\displaystyle &\displaystyle=3m \sqrt[4]{2m^2 } &\mbox{}\\
\end{array}
$$
Square Roots in Their Native Habitat
