Simplifying Square Roots

Using the Product Rule to Simplify Square Roots

To simplify a square root, we rewrite it such that there are no perfect squares in the radicand. There are several properties of square roots that allow us to simplify complicated radical expressions. The first rule we will look at is the product rule for simplifying square roots, which allows us to separate the square root of a product of two numbers into the product of two separate rational expressions. For instance, we can rewrite $\sqrt{15}$ as $\sqrt{3}\cdot \sqrt{5}$ . We can also use the product rule to express the product of multiple radical expressions as a single radical expression.

A General Note: The Product Rule for Simplifying Square Roots

If $a$ and $b$ are nonnegative, the square root of the product $ab$ is equal to the product of the square roots of $a$ and $b$ .

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

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

Factor any perfect squares from the radicand.
Write the radical expression as a product of radical expressions.
Simplify.

Example 2: Using the Product Rule to Simplify Square Roots

Simplify the radical expression.

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

Solution

$\begin{array}{cc}\sqrt{100\cdot 3}\hfill & \text{Factor perfect square from radicand}.\hfill \\ \sqrt{100}\cdot \sqrt{3}\hfill & \text{Write radical expression as product of radical expressions}.\hfill \\ 10\sqrt{3}\hfill & \text{Simplify}.\hfill \\ \text{ }\end{array}$
$\begin{array}{cc}\sqrt{81{a}^{4}{b}^{4}\cdot 2a}\hfill & \text{Factor perfect square from radicand}.\hfill \\ \sqrt{81{a}^{4}{b}^{4}}\cdot \sqrt{2a}\hfill & \text{Write radical expression as product of radical expressions}.\hfill \\ 9{a}^{2}{b}^{2}\sqrt{2a}\hfill & \text{Simplify}.\hfill \end{array}$

Try It 2

Simplify $\sqrt{50{x}^{2}{y}^{3}z}$ .

Solution

How To: Given the product of multiple radical expressions, use the product rule to combine them into one radical expression.

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

Example 3: Using the Product Rule to Simplify the Product of Multiple Square Roots

Simplify the radical expression.

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

Solution

$\begin{array}{cc}\sqrt{12\cdot 3}\hfill & \text{Express the product as a single radical expression}.\hfill \\ \sqrt{36}\hfill & \text{Simplify}.\hfill \\ 6\hfill & \end{array}$

Try It 3

Simplify $\sqrt{50x}\cdot \sqrt{2x}$ assuming $x>0$ .

Solution

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 $\sqrt{\frac{5}{2}}$ as $\frac{\sqrt{5}}{\sqrt{2}}$ .

A General Note: The Quotient Rule for Simplifying Square Roots

The square root of the quotient $\frac{a}{b}$ is equal to the quotient of the square roots of $a$ and $b$ , where $b\ne 0$ .

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

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

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

Example 4: Using the Quotient Rule to Simplify Square Roots

Simplify the radical expression.

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

Solution

$\begin{array}{cc}\frac{\sqrt{5}}{\sqrt{36}}\hfill & \text{Write as quotient of two radical expressions}.\hfill \\ \frac{\sqrt{5}}{6}\hfill & \text{Simplify denominator}.\hfill \end{array}$

Try It 4

Simplify $\sqrt{\frac{2{x}^{2}}{9{y}^{4}}}$ .

Solution

Example 5: Using the Quotient Rule to Simplify an Expression with Two Square Roots

Simplify the radical expression.

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

Solution

$\begin{array}{cc}\sqrt{\frac{234{x}^{11}y}{26{x}^{7}y}}\hfill & \text{Combine numerator and denominator into one radical expression}.\hfill \\ \sqrt{9{x}^{4}}\hfill & \text{Simplify fraction}.\hfill \\ 3{x}^{2}\text{ }\hfill & \text{Simplify square root}.\hfill \end{array}$

Try It 5

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

Solution

Radicals and Rational Expressions

Using the Product Rule to Simplify Square Roots

A General Note: The Product Rule for Simplifying Square Roots

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

Example 2: Using the Product Rule to Simplify Square Roots

Solution

Try It 2

How To: Given the product of multiple radical expressions, use the product rule to combine them into one radical expression.

Example 3: Using the Product Rule to Simplify the Product of Multiple Square Roots

Solution

Try It 3

Using the Quotient Rule to Simplify Square Roots

A General Note: The Quotient Rule for Simplifying Square Roots

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

Example 4: Using the Quotient Rule to Simplify Square Roots

Solution

Try It 4

Example 5: Using the Quotient Rule to Simplify an Expression with Two Square Roots

Solution

Try It 5

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