# 1.3 Radicals and rational expressions  (Page 2/11)

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## The product rule for simplifying square roots

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 $\text{\hspace{0.17em}}b.\text{\hspace{0.17em}}$

$\sqrt{ab}=\sqrt{a}\cdot \sqrt{b}$

Given a square root radical expression, use the product rule to simplify it.

1. Factor any perfect squares from the radicand.
3. Simplify.

## Using the product rule to simplify square roots

1. $\sqrt{300}$
2. $\sqrt{162{a}^{5}{b}^{4}}$

Simplify $\text{\hspace{0.17em}}\sqrt{50{x}^{2}{y}^{3}z}.$

$5|x||y|\sqrt{2yz}.\text{\hspace{0.17em}}$ 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.

1. Express the product of multiple radical expressions as a single radical expression.
2. Simplify.

## Using the product rule to simplify the product of multiple square roots

$\sqrt{12}\cdot \sqrt{3}$

Simplify $\text{\hspace{0.17em}}\sqrt{50x}\cdot \sqrt{2x}\text{\hspace{0.17em}}$ assuming $\text{\hspace{0.17em}}x>0.$

$10|x|$

## Using the quotient rule to simplify square roots

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 quotient rule for simplifying square roots

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 $\text{\hspace{0.17em}}b\ne 0.$

$\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$

Given a radical expression, use the quotient rule to simplify it.

1. Write the radical expression as the quotient of two radical expressions.
2. Simplify the numerator and denominator.

## Using the quotient rule to simplify square roots

$\sqrt{\frac{5}{36}}$

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.

## Using the quotient rule to simplify an expression with two square roots

$\frac{\sqrt{234{x}^{11}y}}{\sqrt{26{x}^{7}y}}$

Simplify $\text{\hspace{0.17em}}\frac{\sqrt{9{a}^{5}{b}^{14}}}{\sqrt{3{a}^{4}{b}^{5}}}.$

${b}^{4}\sqrt{3ab}$

## Adding and subtracting square roots

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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