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If $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ are nonnegative, the square root of the product $\text{\hspace{0.17em}}ab\text{\hspace{0.17em}}$ is equal to the product of the square roots of $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}b.\text{\hspace{0.17em}}$
Given a square root radical expression, use the product rule to simplify it.
Simplify the radical expression.
Simplify $\text{\hspace{0.17em}}\sqrt{50{x}^{2}{y}^{3}z}.$
$5\left|x\right|\left|y\right|\sqrt{2yz}.\text{\hspace{0.17em}}$ Notice the absolute value signs around x and y ? That’s because their value must be positive!
Given the product of multiple radical expressions, use the product rule to combine them into one radical expression.
Simplify the radical expression.
$\sqrt{12}\cdot \sqrt{3}$
$\begin{array}{cc}\sqrt{12\cdot 3}\hfill & \phantom{\rule{2em}{0ex}}\text{Expresstheproductasasingleradicalexpression}.\hfill \\ \sqrt{36}\hfill & \phantom{\rule{2em}{0ex}}\text{Simplify}.\hfill \\ 6\hfill & \end{array}$
Simplify $\text{\hspace{0.17em}}\sqrt{50x}\cdot \sqrt{2x}\text{\hspace{0.17em}}$ assuming $\text{\hspace{0.17em}}x>0.$
$10\left|x\right|$
Just as we can rewrite the square root of a product as a product of square roots, so too can we rewrite the square root of a quotient as a quotient of square roots, using the quotient rule for simplifying square roots. It can be helpful to separate the numerator and denominator of a fraction under a radical so that we can take their square roots separately. We can rewrite $\text{\hspace{0.17em}}\sqrt{\frac{5}{2}}\text{\hspace{0.17em}}$ as $\text{\hspace{0.17em}}\frac{\sqrt{5}}{\sqrt{2}}.$
The square root of the quotient $\text{\hspace{0.17em}}\frac{a}{b}\text{\hspace{0.17em}}$ is equal to the quotient of the square roots of $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}b,$ where $\text{\hspace{0.17em}}b\ne 0.$
Given a radical expression, use the quotient rule to simplify it.
Simplify the radical expression.
$\sqrt{\frac{5}{36}}$
$\begin{array}{cc}\frac{\sqrt{5}}{\sqrt{36}}\hfill & \phantom{\rule{2em}{0ex}}\text{Writeasquotientoftworadicalexpressions}.\hfill \\ \frac{\sqrt{5}}{6}\hfill & \phantom{\rule{2em}{0ex}}\text{Simplifydenominator}.\hfill \end{array}$
Simplify $\text{\hspace{0.17em}}\sqrt{\frac{2{x}^{2}}{9{y}^{4}}}.$
$\frac{x\sqrt{2}}{3{y}^{2}}.\text{\hspace{0.17em}}$ We do not need the absolute value signs for $\text{\hspace{0.17em}}{y}^{2}\text{\hspace{0.17em}}$ because that term will always be nonnegative.
Simplify the radical expression.
$\frac{\sqrt{234{x}^{11}y}}{\sqrt{26{x}^{7}y}}$
$\begin{array}{cc}\sqrt{\frac{234{x}^{11}y}{26{x}^{7}y}}\hfill & \phantom{\rule{2em}{0ex}}\text{Combinenumeratoranddenominatorintooneradicalexpression}.\hfill \\ \sqrt{9{x}^{4}}\hfill & \phantom{\rule{2em}{0ex}}\text{Simplifyfraction}.\hfill \\ 3{x}^{2}\text{}\hfill & \phantom{\rule{2em}{0ex}}\text{Simplifysquareroot}.\hfill \end{array}$
Simplify $\text{\hspace{0.17em}}\frac{\sqrt{9{a}^{5}{b}^{14}}}{\sqrt{3{a}^{4}{b}^{5}}}.$
${b}^{4}\sqrt{3ab}$
We can add or subtract radical expressions only when they have the same radicand and when they have the same radical type such as square roots. For example, the sum of $\text{\hspace{0.17em}}\sqrt{2}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}3\sqrt{2}\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}4\sqrt{2}.\text{\hspace{0.17em}}$ However, it is often possible to simplify radical expressions, and that may change the radicand. The radical expression $\text{\hspace{0.17em}}\sqrt{18}\text{\hspace{0.17em}}$ can be written with a $\text{\hspace{0.17em}}2\text{\hspace{0.17em}}$ in the radicand, as $\text{\hspace{0.17em}}3\sqrt{2},$ so $\text{\hspace{0.17em}}\sqrt{2}+\sqrt{18}=\sqrt{2}+3\sqrt{2}=4\sqrt{2}.$
Given a radical expression requiring addition or subtraction of square roots, solve.
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