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# 5.4: Multiplying and Dividing Radical Expressions

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Skills to Develop

• Rationalize the denominator.

When multiplying radical expressions with the same index, we use the product rule for radicals. Given real numbers $$\sqrt [ n ] { A }$$ and $$\sqrt [ n ] { B }$$,

$$\sqrt [ n ] { A } \cdot \sqrt [ n ] { B } = \sqrt [ n ] { A \cdot B }$$\

Example $$\PageIndex{1}$$:

Multiply: $$\sqrt [ 3 ] { 12 } \cdot \sqrt [ 3 ] { 6 }$$.

Solution:

Apply the product rule for radicals, and then simplify.

\begin{aligned} \sqrt [ 3 ] { 12 } \cdot \sqrt [ 3 ] { 6 } & = \sqrt [ 3 ] { 12 \cdot 6 }\quad \color{Cerulean} { Multiply\: the\: radicands. } \\ & = \sqrt [ 3 ] { 72 } \quad\quad\:\color{Cerulean} { Simplify. } \\ & = \sqrt [ 3 ] { 2 ^ { 3 } \cdot 3 ^ { 2 } } \\ & = 2 \sqrt [ 3 ] { {3 } ^ { 2 }} \\ & = 2 \sqrt [ 3 ] { 9 } \end{aligned}

$$2 \sqrt [ 3 ] { 9 }$$

Often, there will be coefficients in front of the radicals.

Example $$\PageIndex{2}$$:

Multiply: $$3 \sqrt { 6 } \cdot 5 \sqrt { 2 }$$

Solution

Using the product rule for radicals and the fact that multiplication is commutative, we can multiply the coefficients and the radicands as follows.

\begin{aligned} 3 \sqrt { 6 } \cdot 5 \sqrt { 2 } & = \color{Cerulean}{3 \cdot 5}\color{black}{ \cdot}\color{OliveGreen}{ \sqrt { 6 } \cdot \sqrt { 2} }\quad\color{Cerulean}{Multiplication\:is\:commutative.} \\ & = 15 \cdot \sqrt { 12 } \quad\quad\quad\:\color{Cerulean}{Multiply\:the\:coefficients\:and\:the\:radicands.} \\ & = 15 \sqrt { 4 \cdot 3 } \quad\quad\quad\:\color{Cerulean}{Simplify.} \\ & = 15 \cdot 2 \cdot \sqrt { 3 } \\ & = 30 \sqrt { 3 } \end{aligned}

Typically, the first step involving the application of the commutative property is not shown.

$$30 \sqrt { 3 }$$

Example $$\PageIndex{3}$$:

Multiply: $$- 3 \sqrt [ 3 ] { 4 y ^ { 2 } } \cdot 5 \sqrt [ 3 ] { 16 y }$$.

Solution

\begin{aligned} - 3 \sqrt [ 3 ] { 4 y ^ { 2 } } \cdot 5 \sqrt [ 3 ] { 16 y } & = - 15 \sqrt [ 3 ] { 64 y ^ { 3 } }\quad\color{Cerulean}{Multiply\:the\:coefficients\:and\:then\:multipy\:the\:rest.} \\ & = - 15 \sqrt [ 3 ] { 4 ^ { 3 } y ^ { 3 } }\quad\color{Cerulean}{Simplify.} \\ & = - 15 \cdot 4 y \\ & = - 60 y \end{aligned}

$$-60y$$

Use the distributive property when multiplying rational expressions with more than one term.

Example $$\PageIndex{4}$$:

Multiply: $$5 \sqrt { 2 x } ( 3 \sqrt { x } - \sqrt { 2 x } )$$.

Solution:

Apply the distributive property and multiply each term by $$5 \sqrt { 2 x }$$.

\begin{aligned} 5 \sqrt { 2 x } ( 3 \sqrt { x } - \sqrt { 2 x } ) & = \color{Cerulean}{5 \sqrt { 2 x } }\color{black}{\cdot} 3 \sqrt { x } - \color{Cerulean}{5 \sqrt { 2 x }}\color{black}{ \cdot} \sqrt { 2 x } \quad\color{Cerulean}{Distribute.}\\ & = 15 \sqrt { 2 x ^ { 2 } } - 5 \sqrt { 4 x ^ { 2 } } \quad\quad\quad\quad\:\:\:\color{Cerulean}{Simplify.} \\ & = 15 x \sqrt { 2 } - 5 \cdot 2 x \\ & = 15 x \sqrt { 2 } - 10 x \end{aligned}

$$15 x \sqrt { 2 } - 10 x$$

Example $$\PageIndex{5}$$:

Multiply: $$\sqrt [ 3 ] { 6 x ^ { 2 } y } \left( \sqrt [ 3 ] { 9 x ^ { 2 } y ^ { 2 } } - 5 \cdot \sqrt [ 3 ] { 4 x y } \right)$$.

Solution

Apply the distributive property, and then simplify the result.

\begin{aligned} \sqrt [ 3 ] { 6 x ^ { 2 } y } \left( \sqrt [ 3 ] { 9 x ^ { 2 } y ^ { 2 } } - 5 \cdot \sqrt [ 3 ] { 4 x y } \right) & = \color{Cerulean}{\sqrt [ 3 ] { 6 x ^ { 2 } y }}\color{black}{\cdot} \sqrt [ 3 ] { 9 x ^ { 2 } y ^ { 2 } } - \color{Cerulean}{\sqrt [ 3 ] { 6 x ^ { 2 } y }}\color{black}{ \cdot} 5 \sqrt [ 3 ] { 4 x y } \\ & = \sqrt [ 3 ] { 54 x ^ { 4 } y ^ { 3 } } - 5 \sqrt [ 3 ] { 24 x ^ { 3 } y ^ { 2 } } \\ & = \sqrt [ 3 ] { 27 \cdot 2 \cdot x \cdot x ^ { 3 } \cdot y ^ { 3 } } - 5 \sqrt [ 3 ] { 8 \cdot 3 \cdot x ^ { 3 } \cdot y ^ { 2 } } \\ & = 3 x y \sqrt [ 3 ] { 2 x } - 10 x \sqrt [ 3 ] { 3 y ^ { 2 } } \\ & = 3 x y \sqrt [ 3 ] { 2 x } - 10 x \sqrt [ 3 ] { 3 y ^ { 2 } } \end{aligned}

$$3 x y \sqrt [ 3 ] { 2 x } - 10 x \sqrt [ 3 ] { 3 y ^ { 2 } }$$

The process for multiplying radical expressions with multiple terms is the same process used when multiplying polynomials. Apply the distributive property, simplify each radical, and then combine like terms.

Example $$\PageIndex{6}$$:

Multiply: $$( \sqrt { x } - 5 \sqrt { y } ) ^ { 2 }$$.

Solution

$$( \sqrt { x } - 5 \sqrt { y } ) ^ { 2 } = ( \sqrt { x } - 5 \sqrt { y } ) ( \sqrt { x } - 5 \sqrt { y } )$$

Begin by applying the distributive property.

$$\begin{array} { l } { = \color{Cerulean}{\sqrt { x }}\color{black}{ \cdot} \sqrt { x } + \color{Cerulean}{\sqrt { x }}\color{black}{ (} - 5 \sqrt { y } ) + ( \color{OliveGreen}{- 5 \sqrt { y }}\color{black}{ )} \sqrt { x } + ( \color{OliveGreen}{- 5 \sqrt { y }}\color{black}{ )} ( - 5 \sqrt { y } ) } \\ { = \sqrt { x ^ { 2 } } - 5 \sqrt { x y } - 5 \sqrt { x y } + 25 \sqrt { y ^ { 2 } } } \\ { = x - 10 \sqrt { x y } + 25 y } \end{array}$$

$$x - 10 \sqrt { x y } + 25 y$$

The binomials $$(a + b)$$ and $$(a − b)$$ are called conjugates18. When multiplying conjugate binomials the middle terms are opposites and their sum is zero.

Example $$\PageIndex{7}$$:

Multiply: $$( \sqrt { 10 } + \sqrt { 3 } ) ( \sqrt { 10 } - \sqrt { 3 } )$$.

Solution

Apply the distributive property, and then combine like terms.

\begin{aligned} ( \sqrt { 10 } + \sqrt { 3 } ) ( \sqrt { 10 } - \sqrt { 3 } ) & = \color{Cerulean}{\sqrt { 10} }\color{black}{ \cdot} \sqrt { 10 } + \color{Cerulean}{\sqrt { 10} }\color{black}{ (} - \sqrt { 3 } ) + \color{OliveGreen}{\sqrt{3}}\color{black}{ (}\sqrt{10}) + \color{OliveGreen}{\sqrt{3}}\color{black}{(}-\sqrt{3}) \\ & = \sqrt { 100 } - \sqrt { 30 } + \sqrt { 30 } - \sqrt { 9 } \\ & = 10 - \color{red}{\sqrt { 30 }}\color{black}{ +}\color{red}{ \sqrt { 30} }\color{black}{ -} 3 \\ & = 10 - 3 \\ & = 7 \\ \end{aligned}

$$7$$

It is important to note that when multiplying conjugate radical expressions, we obtain a rational expression. This is true in general

\begin{aligned} ( \sqrt { x } + \sqrt { y } ) ( \sqrt { x } - \sqrt { y } ) & = \sqrt { x ^ { 2 } } - \sqrt { x y } + \sqrt {x y } - \sqrt { y ^ { 2 } } \\ & = x - y \end{aligned}

Alternatively, using the formula for the difference of squares we have,

\begin{aligned} ( a + b ) ( a - b ) & = a ^ { 2 } - b ^ { 2 }\quad\quad\quad\color{Cerulean}{Difference\:of\:squares.} \\ ( \sqrt { x } + \sqrt { y } ) ( \sqrt { x } - \sqrt { y } ) & = ( \sqrt { x } ) ^ { 2 } - ( \sqrt { y } ) ^ { 2 } \\ & = x - y \end{aligned}

Exercise $$\PageIndex{1}$$

Multiply: $$( 3 - 2 \sqrt { y } ) ( 3 + 2 \sqrt { y } )$$. (Assume $$y$$ is positive.)

$$9-4y$$

To divide radical expressions with the same index, we use the quotient rule for radicals. Given real numbers $$\sqrt [ n ] { A }$$ and $$\sqrt [ n ] { B }$$,

$$\frac { \sqrt [ n ] { A } } { \sqrt [ n ] { B } } = \sqrt [n]{ \frac { A } { B } }$$

Example $$\PageIndex{8}$$:

Divide: $$\frac { \sqrt [ 3 ] { 96 } } { \sqrt [ 3 ] { 6 } }$$.

Solution

In this case, we can see that $$6$$ and $$96$$ have common factors. If we apply the quotient rule for radicals and write it as a single cube root, we will be able to reduce the fractional radicand.

\begin{aligned} \frac { \sqrt [ 3 ] { 96 } } { \sqrt [ 3 ] { 6 } } & = \sqrt [ 3 ] { \frac { 96 } { 6 } } \quad\color{Cerulean}{Apply\:the\:quotient\:rule\:for\:radicals\:and\:reduce\:the\:radicand.}\\ & = \sqrt [ 3 ] { 16 } \\ & = \sqrt [ 3 ] { 8 \cdot 2 } \color{Cerulean}{Simplify.} \\ & = 2 \sqrt [ 3 ] { 2 } \end{aligned}

$$2 \sqrt [ 3 ] { 2 }$$

Example $$\PageIndex{9}$$:

Divide: $$\frac { \sqrt { 50 x ^ { 6 } y ^ { 4} } } { \sqrt { 8 x ^ { 3 } y } }$$.

Solution

Write as a single square root and cancel common factors before simplifying.

\begin{aligned} \frac { \sqrt { 50 x ^ { 6 } y ^ { 4 } } } { \sqrt { 8 x ^ { 3 } y } } & = \sqrt { \frac { 50 x ^ { 6 } y ^ { 4 } } { 8 x ^ { 3 } y } } \quad\color{Cerulean}{Apply\:the\:quotient\:rule\:for\:radicals\:and\:cancel.}\\ & = \sqrt { \frac { 25 x ^ { 3 } y ^ { 3 } } { 4 } } \quad\color{Cerulean}{Simplify.} \\ & = \frac { \sqrt { 25 x ^ { 3 } y ^ { 3 } } } { \sqrt { 4 } } \\ & = \frac { 5 x y \sqrt { x y } } { 2 } \end{aligned}

$$\frac { 5xy \sqrt {x y } } { 2 }$$

## Rationalizing the Denominator

When the denominator (divisor) of a radical expression contains a radical, it is a common practice to find an equivalent expression where the denominator is a rational number. Finding such an equivalent expression is called rationalizing the denominator19.

$$\begin{array} { c } { \color{Cerulean} { Radical\:expression\quad Rational\: denominator } } \\ { \frac { 1 } { \sqrt { 2 } } \quad\quad\quad=\quad\quad\quad\quad \frac { \sqrt { 2 } } { 2 } } \end{array}$$

To do this, multiply the fraction by a special form of $$1$$ so that the radicand in the denominator can be written with a power that matches the index. After doing this, simplify and eliminate the radical in the denominator. For example:

$$\frac { 1 } { \sqrt { 2 } } = \frac { 1 } { \sqrt { 2 } } \cdot \frac { \color{Cerulean}{\sqrt { 2} } } {\color{Cerulean}{ \sqrt { 2} } } \color{black}{=} \frac { \sqrt { 2 } } { \sqrt { 4 } } = \frac { \sqrt { 2 } } { 2 }$$

Remember, to obtain an equivalent expression, you must multiply the numerator and denominator by the exact same nonzero factor.

Example $$\PageIndex{10}$$:

Rationalize the denominator: $$\frac { \sqrt { 2 } } { \sqrt { 5 x } }$$.

Solution

The goal is to find an equivalent expression without a radical in the denominator. The radicand in the denominator determines the factors that you need to use to rationalize it. In this example, multiply by $$1$$ in the form $$\frac { \sqrt { 5 x } } { \sqrt { 5 x } }$$.

\begin{aligned} \frac { \sqrt { 2 } } { \sqrt { 5 x } } & = \frac { \sqrt { 2 } } { \sqrt { 5 x } } \cdot \color{Cerulean}{\frac { \sqrt { 5 x } } { \sqrt { 5 x } } { \:Multiply\:by\: } \frac { \sqrt { 5 x } } { \sqrt { 5 x } } .}\\ & = \frac { \sqrt { 10 x } } { \sqrt { 25 x ^ { 2 } } } \quad\quad\: \color{Cerulean} { Simplify. } \\ & = \frac { \sqrt { 10 x } } { 5 x } \end{aligned}

$$\frac { \sqrt { 10 x } } { 5 x }$$

Sometimes, we will find the need to reduce, or cancel, after rationalizing the denominator.

Example $$\PageIndex{11}$$:

Rationalize the denominator: $$\frac { 3 a \sqrt { 2 } } { \sqrt { 6 a b } }$$.

Solution

In this example, we will multiply by $$1$$ in the form $$\frac { \sqrt { 6 a b } } { \sqrt { 6 a b } }$$.

\begin{aligned} \frac { 3 a \sqrt { 2 } } { \sqrt { 6 a b } } & = \frac { 3 a \sqrt { 2 } } { \sqrt { 6 a b } } \cdot \color{Cerulean}{\frac { \sqrt { 6 a b } } { \sqrt { 6 a b } }} \\ & = \frac { 3 a \sqrt { 12 a b } } { \sqrt { 36 a ^ { 2 } b ^ { 2 } } } \quad\quad\color{Cerulean}{Simplify.}\\ & = \frac { 3 a \sqrt { 4 \cdot 3 a b} } { 6 ab } \\ & = \frac { 6 a \sqrt { 3 a b } } { b }\quad\quad\:\:\color{Cerulean}{Cancel.} \\ & = \frac { \sqrt { 3 a b } } { b } \end{aligned}

Notice that $$b$$ does not cancel in this example. Do not cancel factors inside a radical with those that are outside.

$$\frac { \sqrt { 3 a b } } { b }$$

Exercise $$\PageIndex{2}$$

Rationalize the denominator: $$\sqrt { \frac { 9 x } { 2 y } }$$.

$$\frac { 3 \sqrt { 2xy } } { 2 y }$$

Up to this point, we have seen that multiplying a numerator and a denominator by a square root with the exact same radicand results in a rational denominator. In general, this is true only when the denominator contains a square root. However, this is not the case for a cube root. For example,

$$\frac { 1 } { \sqrt [ 3 ] { x } } \cdot \color{Cerulean}{\frac { \sqrt [ 3 ] { x } } { \sqrt [ 3 ] { x } }}\color{black}{ =} \frac { \sqrt [ 3 ] { x } } { \sqrt [ 3 ] { x ^ { 2 } } }$$

Note that multiplying by the same factor in the denominator does not rationalize it. In this case, if we multiply by $$1$$ in the form of $$\frac { \sqrt [ 3 ] { x ^ { 2 } } } { \sqrt [ 3 ] { x ^ { 2 } } }$$, then we can write the radicand in the denominator as a power of $$3$$. Simplifying the result then yields a rationalized denominator.

$$\frac { 1 } { \sqrt [ 3 ] { x } } = \frac { 1 } { \sqrt [ 3 ] { x } } \cdot \color{Cerulean}{\frac { \sqrt [ 3 ] { x ^ { 2 } } } { \sqrt [ 3 ] { x ^ { 2 } } }} = \frac { \sqrt [ 3 ] { x ^ { 2 } } } { \sqrt [ 3 ] { x ^ { 3 } } } = \frac { \sqrt [ 3 ] { x ^ { 2 } } } { x }$$

Therefore, to rationalize the denominator of a radical expression with one radical term in the denominator, begin by factoring the radicand of the denominator. The factors of this radicand and the index determine what we should multiply by. Multiply the numerator and denominator by the $$n$$th root of factors that produce nth powers of all the factors in the radicand of the denominator.

Example $$\PageIndex{12}$$:

Rationalize the denominator: $$\frac { \sqrt [ 3 ] { 2 } } { \sqrt [ 3 ] { 25 } }$$.

Solution

The radical in the denominator is equivalent to $$\sqrt [ 3 ] { 5 ^ { 2 } }$$. To rationalize the denominator, we need: $$\sqrt [ 3 ] { 5 ^ { 3 } }$$. To obtain this, we need one more factor of $$5$$. Therefore, multiply by $$1$$ in the form of $$\frac { \sqrt [3]{ 5 } } { \sqrt[3] { 5 } }$$.

\begin{aligned} \frac { \sqrt [ 3 ] { 2 } } { \sqrt [ 3 ] { 25 } } & = \frac { \sqrt [ 3 ] { 2 } } { \sqrt [ 3 ] { 5 ^ { 2 } } } \cdot \color{Cerulean}{\frac { \sqrt [ 3 ] { 5 } } { \sqrt [ 3 ] { 5 } } \:Multiply\:by\:the\:cube\:root\:of\:factors\:that\:result\:in\:powers\:of\:3.} \\ & = \frac { \sqrt [ 3 ] { 10 } } { \sqrt [ 3 ] { 5 ^ { 3 } } } \quad\:\:\:\quad\color{Cerulean}{Simplify.} \\ & = \frac { \sqrt [ 3 ] { 10 } } { 5 } \end{aligned}

$$\frac { \sqrt [ 3 ] { 10 } } { 5 }$$

Example $$\PageIndex{13}$$:

Rationalize the denominator: $$\sqrt [ 3 ] { \frac { 27 a } { 2 b ^ { 2 } } }$$.

Solution

In this example, we will multiply by $$1$$ in the form $$\frac { \sqrt [ 3 ] { 2 ^ { 2 } b } } { \sqrt [ 3 ] { 2 ^ { 2 } b } }$$.

\begin{aligned} \sqrt [ 3 ] { \frac { 27 a } { 2 b ^ { 2 } } } & = \frac { \sqrt [ 3 ] { 3 ^ { 3 } a } } { \sqrt [ 3 ] { 2 b ^ { 2 } } } \quad\quad\quad\quad\color{Cerulean}{Apply\:the\:quotient\:rule\:for\:radicals.} \\ & = \frac { 3 \sqrt [ 3 ] { a } } { \sqrt [ 3 ] { 2 b ^ { 2 } } } \cdot \color{Cerulean}{\frac { \sqrt [ 3 ] { 2 ^ { 2 } b } } { \sqrt [ 3 ] { 2 ^ { 2 } b } }\:\:\:Multiply\:by\:the\:cube\:root\:of\:factors\:that\:result\:in\:powers.} \\ & = \frac { 3 \sqrt [ 3 ] { 2 ^ { 2 } ab } } { \sqrt [ 3 ] { 2 ^ { 3 } b ^ { 3 } } } \quad\quad\quad\color{Cerulean}{Simplify.}\\ & = \frac { 3 \sqrt [ 3 ] { 4 a b } } { 2 b } \end{aligned}

$$\frac { 3 \sqrt [ 3 ] { 4 a b } } { 2 b }$$

Example $$\PageIndex{14}$$:

Rationalize the denominator: $$\frac { 2 x \sqrt [ 5 ] { 5 } } { \sqrt [ 5 ] { 4 x ^ { 3 } y } }$$

Solution

In this example, we will multiply by $$1$$ in the form $$\frac { \sqrt [ 5 ] { 2 ^ { 3 } x ^ { 2 } y ^ { 4 } } } { \sqrt [ 5 ] { 2 ^ { 3 } x ^ { 2 } y ^ { 4 } } }$$

\begin{aligned} \frac{2x\sqrt[5]{5}}{\sqrt[5]{4x^{3}y}} & = \frac{2x\sqrt[5]{5}}{\sqrt[5]{2^{2}x^{3}y}}\cdot\color{Cerulean}{\frac{\sqrt[5]{2^{3}x^{2}y^{4}}}{\sqrt[5]{2^{3}x^{2}y^{4}}} \:\:Multiply\:by\:the\:fifth\:root\:of\:factors\:that\:result\:in\:pairs.} \\ & = \frac { 2 x \sqrt [ 5 ] { 5 \cdot 2 ^ { 3 } x ^ { 2 } y ^ { 4 } } } { \sqrt [ 5 ] { 2 ^ { 5 } x ^ { 5 } y ^ { 5 } } } \quad\quad\:\:\color{Cerulean}{Simplify.} \\ & = \frac { 2 x \sqrt [ 5 ] { 40 x ^ { 2 } y ^ { 4 } } } { 2 x y } \\ & = \frac { \sqrt [ 5 ] { 40 x ^ { 2 } y ^ { 4 } } } { y } \end{aligned}

$$\frac { \sqrt [ 5 ] { 40 x ^ { 2 } y ^ { 4 } } } { y }$$

When two terms involving square roots appear in the denominator, we can rationalize it using a very special technique. This technique involves multiplying the numerator and the denominator of the fraction by the conjugate of the denominator. Recall that multiplying a radical expression by its conjugate produces a rational number.

Example $$\PageIndex{15}$$:

Rationalize the denominator: $$\frac { 1 } { \sqrt { 5 } - \sqrt { 3 } }$$.

Solution

In this example, the conjugate of the denominator is $$\sqrt { 5 } + \sqrt { 3 }$$. Therefore, multiply by $$1$$ in the form $$\frac { ( \sqrt { 5 } + \sqrt { 3 } ) } { ( \sqrt {5 } + \sqrt { 3 } ) }$$.

\begin{aligned} \frac { 1 } { \sqrt { 5 } - \sqrt { 3 } } & = \frac { 1 } { ( \sqrt { 5 } - \sqrt { 3 } ) } \color{Cerulean}{\frac { ( \sqrt { 5 } + \sqrt { 3 } ) } { ( \sqrt { 5 } + \sqrt { 3 } ) } \:\:Multiply \:numerator\:and\:denominator\:by\:the\:conjugate\:of\:the\:denominator.} \\ & = \frac { \sqrt { 5 } + \sqrt { 3 } } { \sqrt { 25 } + \sqrt { 15 } - \sqrt{15}-\sqrt{9} } \:\color{Cerulean}{Simplify.} \\ & = \frac { \sqrt { 5 } + \sqrt { 3 } } { 5-3 } \\ & = \frac { \sqrt { 5 } + \sqrt { 3 } } { 2 } \end{aligned}

$$\frac { \sqrt { 5 } + \sqrt { 3 } } { 2 }$$

Notice that the terms involving the square root in the denominator are eliminated by multiplying by the conjugate. We can use the property $$( \sqrt { a } + \sqrt { b } ) ( \sqrt { a } - \sqrt { b } ) = a - b$$ to expedite the process of multiplying the expressions in the denominator.

Example $$\PageIndex{16}$$:

Rationalize the denominator: $$\frac { \sqrt { 10 } } { \sqrt { 2 } + \sqrt { 6 } }$$.

Solution

Multiply by $$1$$ in the form $$\frac { \sqrt { 2 } - \sqrt { 6 } } { \sqrt { 2 } - \sqrt { 6 } }$$.

\begin{aligned} \frac{\sqrt{10}}{\sqrt{2}+\sqrt{6} }&= \frac{(\sqrt{10})}{(\sqrt{2}+\sqrt{6})} \color{Cerulean}{\frac{(\sqrt{2}-\sqrt{6})}{(\sqrt{2}-\sqrt{6})}\quad\quad Multiple\:by\:the\:conjugate.} \\ &= \frac { \sqrt { 20 } - \sqrt { 60 } } { 2 - 6 } \quad\quad\quad\quad\quad\quad\:\:\:\color{Cerulean}{Simplify.} \\ &= \frac { \sqrt { 4 \cdot 5 } - \sqrt { 4 \cdot 15 } } { - 4 } \\ &= \frac { 2 \sqrt { 5 } - 2 \sqrt { 15 } } { - 4 } \\ &=\frac{2(\sqrt{5}-\sqrt{15})}{-4} \\ &= \frac { \sqrt { 5 } - \sqrt { 15 } } { - 2 } = - \frac { \sqrt { 5 } - \sqrt { 15 } } { 2 } = \frac { - \sqrt { 5 } + \sqrt { 15 } } { 2 } \end{aligned}

$$\frac { \sqrt { 15 } - \sqrt { 5 } } { 2 }$$

Example $$\PageIndex{17}$$:

Rationalize the denominator: $$\frac { \sqrt { x } - \sqrt { y } } { \sqrt { x } + \sqrt { y } }$$.

Solution

In this example, we will multiply by $$1$$ in the form $$\frac { \sqrt { x } - \sqrt { y } } { \sqrt { x } - \sqrt { y } }$$.

\begin{aligned} \frac { \sqrt { x } - \sqrt { y } } { \sqrt { x } + \sqrt { y } } & = \frac { ( \sqrt { x } - \sqrt { y } ) } { ( \sqrt { x } + \sqrt { y } ) } \color{Cerulean}{\frac { ( \sqrt { x } - \sqrt { y } ) } { ( \sqrt { x } - \sqrt { y } ) } \quad \quad Multiply\:by\:the\:conjugate\:of\:the\:denominator.} \\ & = \frac { \sqrt { x ^ { 2 } } - \sqrt { x y } - \sqrt { x y } + \sqrt { y ^ { 2 } } } { x - y } \:\:\color{Cerulean}{Simplify.} \\ & = \frac { x - 2 \sqrt { x y } + y } { x - y } \end{aligned}

$$\frac { x - 2 \sqrt { x y } + y } { x - y }$$

Exercise $$\PageIndex{3}$$

Rationalize the denominator: $$\frac { 2 \sqrt { 3 } } { 5 - \sqrt { 3 } }$$

$$\frac { 5 \sqrt { 3 } + 3 } { 11 }$$

## Key Takeaways

• To multiply two single-term radical expressions, multiply the coefficients and multiply the radicands. If possible, simplify the result.
• Apply the distributive property when multiplying a radical expression with multiple terms. Then simplify and combine all like radicals.
• Multiplying a two-term radical expression involving square roots by its conjugate results in a rational expression.
• It is common practice to write radical expressions without radicals in the denominator. The process of finding such an equivalent expression is called rationalizing the denominator.
• If an expression has one term in the denominator involving a radical, then rationalize it by multiplying the numerator and denominator by the $$n$$th root of factors of the radicand so that their powers equal the index.
• If a radical expression has two terms in the denominator involving square roots, then rationalize it by multiplying the numerator and denominator by the conjugate of the denominator.

Exercise $$\PageIndex{4}$$

Multiply. (Assume all variables represent non-negative real numbers.)

1. $$\sqrt { 3 } \cdot \sqrt { 7 }$$
2. $$\sqrt { 2 } \cdot \sqrt { 5 }$$
3. $$\sqrt { 6 } \cdot \sqrt { 12 }$$
4. $$\sqrt { 10 } \cdot \sqrt { 15 }$$
5. $$\sqrt { 2 } \cdot \sqrt { 6 }$$
6. $$\sqrt { 5 } \cdot \sqrt { 15 }$$
7. $$\sqrt { 7 } \cdot \sqrt { 7 }$$
8. $$\sqrt { 12 } \cdot \sqrt { 12 }$$
9. $$2 \sqrt { 5 } \cdot 7 \sqrt { 10 }$$
10. $$3 \sqrt { 15 } \cdot 2 \sqrt { 6 }$$
11. $$( 2 \sqrt { 5 } ) ^ { 2 }$$
12. $$( 6 \sqrt { 2 } ) ^ { 2 }$$
13. $$\sqrt { 2 x } \cdot \sqrt { 2 x }$$
14. $$\sqrt { 5 y } \cdot \sqrt { 5 y }$$
15. $$\sqrt { 3 a } \cdot \sqrt { 12 }$$
16. $$\sqrt { 3 a } \cdot \sqrt { 2 a }$$
17. $$4 \sqrt { 2 x } \cdot 3 \sqrt { 6 x }$$
18. $$5 \sqrt { 10 y } \cdot 2 \sqrt { 2 y }$$
19. $$\sqrt [ 3 ] { 3 } \cdot \sqrt [ 3 ] { 9 }$$
20. $$\sqrt [ 3 ] { 4 } \cdot \sqrt [ 3 ] { 16 }$$
21. $$\sqrt [ 3 ] { 15 } \cdot \sqrt [ 3 ] { 25 }$$
22. $$\sqrt [ 3 ] { 100 } \cdot \sqrt [ 3 ] { 50 }$$
23. $$\sqrt [ 3 ] { 4 } \cdot \sqrt [ 3 ] { 10 }$$
24. $$\sqrt [ 3 ] { 18 } \cdot \sqrt [ 3 ] { 6 }$$
25. $$( 5 \sqrt [ 3 ] { 9 } ) ( 2 \sqrt [ 3 ] { 6 } )$$
26. $$( 2 \sqrt [ 3 ] { 4 } ) ( 3 \sqrt [ 3 ] { 4 } )$$
27. $$( 2 \sqrt [ 3 ] { 2 } ) ^ { 3 }$$
28. $$( 3 \sqrt [ 3 ] { 4 } ) ^ { 3 }$$
29. $$\sqrt [ 3 ] { 3 a ^ { 2 } } \cdot \sqrt [ 3 ] { 9 a }$$
30. $$\sqrt [ 3 ] { 7 b } \cdot \sqrt [ 3 ] { 49 b ^ { 2 } }$$
31. $$\sqrt [ 3 ] { 6 x ^ { 2 } } \cdot \sqrt [ 3 ] { 4 x ^ { 2 } }$$
32. $$\sqrt [ 3 ] { 12 y } \cdot \sqrt [ 3 ] { 9 y ^ { 2 } }$$
33. $$\sqrt [ 3 ] { 20 x ^ { 2 } y } \cdot \sqrt [ 3 ] { 10 x ^ { 2 } y ^ { 2 } }$$
34. $$\sqrt [ 3 ] { 63 x y } \cdot \sqrt [ 3 ] { 12 x ^ { 4 } y ^ { 2 } }$$
35. $$\sqrt { 5 } ( 3 - \sqrt { 5 } )$$
36. $$\sqrt { 2 } ( \sqrt { 3 } - \sqrt { 2 } )$$
37. $$3 \sqrt { 7 } ( 2 \sqrt { 7 } - \sqrt { 3 } )$$
38. $$2 \sqrt { 5 } ( 6 - 3 \sqrt { 10 } )$$
39. $$\sqrt { 6 } ( \sqrt { 3 } - \sqrt { 2 } )$$
40. $$\sqrt { 15 } ( \sqrt { 5 } + \sqrt { 3 } )$$
41. $$\sqrt { x } ( \sqrt { x } + \sqrt { x y } )$$
42. $$\sqrt { y } ( \sqrt { x y } + \sqrt { y } )$$
43. $$\sqrt { 2 a b } ( \sqrt { 14 a } - 2 \sqrt { 10 b } )$$
44. $$\sqrt { 6 a b } ( 5 \sqrt { 2 a } - \sqrt { 3 b } )$$
45. $$\sqrt [ 3 ] { 6 } ( \sqrt [ 3 ] { 9 } - \sqrt [ 3 ] { 20 } )$$
46. $$\sqrt [ 3 ] { 12 } ( \sqrt [ 3 ] { 36 } + \sqrt [ 3 ] { 14 } )$$
47. $$( \sqrt { 2 } - \sqrt { 5 } ) ( \sqrt { 3 } + \sqrt { 7 } )$$
48. $$( \sqrt { 3 } + \sqrt { 2 } ) ( \sqrt { 5 } - \sqrt { 7 } )$$
49. $$( 2 \sqrt { 3 } - 4 ) ( 3 \sqrt { 6 } + 1 )$$
50. $$( 5 - 2 \sqrt { 6 } ) ( 7 - 2 \sqrt { 3 } )$$
51. $$( \sqrt { 5 } - \sqrt { 3 } ) ^ { 2 }$$
52. $$( \sqrt { 7 } - \sqrt { 2 } ) ^ { 2 }$$
53. $$( 2 \sqrt { 3 } + \sqrt { 2 } ) ( 2 \sqrt { 3 } - \sqrt { 2 } )$$
54. $$( \sqrt { 2 } + 3 \sqrt { 7 } ) ( \sqrt { 2 } - 3 \sqrt { 7 } )$$
55. $$( \sqrt { a } - \sqrt { 2 b } ) ^ { 2 }$$
56. $$( \sqrt { a b } + 1 ) ^ { 2 }$$
57. What is the perimeter and area of a rectangle with length measuring $$5\sqrt{3}$$ centimeters and width measuring $$3\sqrt{2}$$ centimeters?
58. What is the perimeter and area of a rectangle with length measuring $$2\sqrt{6}$$ centimeters and width measuring $$\sqrt{3}$$ centimeters?
59. If the base of a triangle measures $$6\sqrt{2}$$ meters and the height measures $$3\sqrt{2}$$ meters, then calculate the area.
60. If the base of a triangle measures $$6\sqrt{3}$$ meters and the height measures $$3\sqrt{6}$$ meters, then calculate the area.

1. $$\sqrt{21}$$

3. $$6\sqrt{2}$$

5. $$2\sqrt{3}$$

7. $$7$$

9. $$70\sqrt{2}$$

11. $$20$$

13. $$2x$$

15. $$6\sqrt{a}$$

17. $$24x\sqrt{3}$$

19. $$3$$

21. $$5 \sqrt [ 3 ] { 3 }$$

23. $$2 \sqrt [ 3 ] { 5 }$$

25. $$30 \sqrt [ 3 ] { 2 }$$

27. $$16$$

29. $$3a$$

31. $$2 x \sqrt [ 3 ] { 3 x }$$

33. $$2 x y \sqrt [ 3 ] { 25 x }$$

35. $$3\sqrt{5}-5$$

37. $$42 - 3 \sqrt { 21 }$$

39. $$3 \sqrt { 2 } - 2 \sqrt { 3 }$$

41. $$x + x \sqrt { y }$$

43. $$2 a \sqrt { 7 b } - 4 b \sqrt { 5 a }$$

45. $$3 \sqrt [ 3 ] { 2 } - 2 \sqrt [ 3 ] { 15 }$$

47. $$\sqrt { 6 } + \sqrt { 14 } - \sqrt { 15 } - \sqrt { 35 }$$

49. $$18 \sqrt { 2 } + 2 \sqrt { 3 } - 12 \sqrt { 6 } - 4$$

51. $$8 - 2 \sqrt { 15 }$$

53. $$10$$

55. $$a - 2 \sqrt { 2 a b } + 2 b$$

57. Perimeter: $$( 10 \sqrt { 3 } + 6 \sqrt { 2 } )$$ centimeters; area $$15\sqrt{6}$$ square centimeters

59. $$18$$ square meters

Exercise $$\PageIndex{5}$$

Divide. (Assume all variables represent positive real numbers.)

1. $$\frac { \sqrt { 75 } } { \sqrt { 3 } }$$
2. $$\frac { \sqrt { 360 } } { \sqrt { 10 } }$$
3. $$\frac { \sqrt { 72 } } { \sqrt { 75 } }$$
4. $$\frac { \sqrt { 90 } } { \sqrt { 98 } }$$
5. $$\frac { \sqrt { 90 x ^ { 5 } } } { \sqrt { 2 x } }$$
6. $$\frac { \sqrt { 96 y ^ { 3 } } } { \sqrt { 3 y } }$$
7. $$\frac { \sqrt { 162 x ^ { 7 } y ^ { 5 } } } { \sqrt { 2 x y } }$$
8. $$\frac { \sqrt { 363 x ^ { 4 } y ^ { 9 } } } { \sqrt { 3 x y } }$$
9. $$\frac { \sqrt [ 3 ] { 16 a ^ { 5 } b ^ { 2 } } } { \sqrt [ 3 ] { 2 a ^ { 2 } b ^ { 2 } } }$$
10. $$\frac { \sqrt [ 3 ] { 192 a ^ { 2 } b ^ { 7 } } } { \sqrt [ 3 ] { 2 a ^ { 2 } b ^ { 2 } } }$$

1. $$5$$

3. $$\frac { 2 \sqrt { 6 } } { 5 }$$

5. $$3 x ^ { 2 } \sqrt { 5 }$$

7. $$9 x ^ { 3 } y ^ { 2 }$$

9. $$2a$$

Exercise $$\PageIndex{6}$$

Rationalize the denominator. (Assume all variables represent positive real numbers.)

1. $$\frac { 1 } { \sqrt { 5 } }$$
2. $$\frac { 1 } { \sqrt { 6 } }$$
3. $$\frac { \sqrt { 2 } } { \sqrt { 3 } }$$
4. $$\frac { \sqrt { 3 } } { \sqrt { 7 } }$$
5. $$\frac { 5 } { 2 \sqrt { 10 } }$$
6. $$\frac { 3 } { 5 \sqrt { 6 } }$$
7. $$\frac { \sqrt { 3 } - \sqrt { 5 } } { \sqrt { 3 } }$$
8. $$\frac { \sqrt { 6 } - \sqrt { 2 } } { \sqrt { 2 } }$$
9. $$\frac { 1 } { \sqrt { 7 x } }$$
10. $$\frac { 1 } { \sqrt { 3 y } }$$
11. $$\frac { a } { 5 \sqrt { a b } }$$
12. $$\frac { 3 b ^ { 2 } } { 2 \sqrt { 3 a b } }$$
13. $$\frac { 2 } { \sqrt [ 3 ] { 36 } }$$
14. $$\frac { 14 } { \sqrt [ 3 ] { 7 } }$$
15. $$\frac { 1 } { \sqrt [ 3 ] { 4 x } }$$
16. $$\frac { 1 } { \sqrt [ 3 ] { 3 y ^ { 2 } } }$$
17. $$\frac { 9 x \sqrt[3] { 2 } } { \sqrt [ 3 ] { 9 x y ^ { 2 } } }$$
18. $$\frac { 5 y ^ { 2 } \sqrt [ 3 ] { x } } { \sqrt [ 3 ] { 5 x ^ { 2 } y } }$$
19. $$\frac { 3 a } { 2 \sqrt [ 3 ] { 3 a ^ { 2 } b ^ { 2 } } }$$
20. $$\frac { 25 n } { 3 \sqrt [ 3 ] { 25 m ^ { 2 } n } }$$
21. $$\frac { 3 } { \sqrt [ 5 ] { 27 x ^ { 2 } y } }$$
22. $$\frac { 2 } { \sqrt [ 5 ] { 16 x y ^ { 2 } } }$$
23. $$\frac { a b } { \sqrt [ 5 ] { 9 a ^ { 3 } b } }$$
24. $$\frac { a b c } { \sqrt [ 5 ] { a b ^ { 2 } c ^ { 3 } } }$$
25. $$\sqrt [ 5 ] { \frac { 3 x } { 8 y ^ { 2 } z } }$$
26. $$\sqrt [ 5 ] { \frac { 4 x y ^ { 2 } } { 9 x ^ { 3 } y z ^ { 4 } } }$$
27. $$\frac { 3 } { \sqrt { 10 } - 3 }$$
28. $$\frac { 2 } { \sqrt { 6 } - 2 }$$
29. $$\frac { 1 } { \sqrt { 5 } + \sqrt { 3 } }$$
30. $$\frac { 1 } { \sqrt { 7 } - \sqrt { 2 } }$$
31. $$\frac { \sqrt { 3 } } { \sqrt { 3 } + \sqrt { 6 } }$$
32. $$\frac { \sqrt { 5 } } { \sqrt { 5 } + \sqrt { 15 } }$$
33. $$\frac { 10 } { 5 - 3 \sqrt { 5 } }$$
34. $$\frac { - 2 \sqrt { 2 } } { 4 - 3 \sqrt { 2 } }$$
35. $$\frac { \sqrt { 3 } + \sqrt { 5 } } { \sqrt { 3 } - \sqrt { 5 } }$$
36. $$\frac { \sqrt { 10 } - \sqrt { 2 } } { \sqrt { 10 } + \sqrt { 2 } }$$
37. $$\frac { 2 \sqrt { 3 } - 3 \sqrt { 2 } } { 4 \sqrt { 3 } + \sqrt { 2 } }$$
38. $$\frac { 6 \sqrt { 5 } + 2 } { 2 \sqrt { 5 } - \sqrt { 2 } }$$
39. $$\frac { x - y } { \sqrt { x } + \sqrt { y } }$$
40. $$\frac { x - y } { \sqrt { x } - \sqrt { y } }$$
41. $$\frac { x + \sqrt { y } } { x - \sqrt { y } }$$
42. $$\frac { x - \sqrt { y } } { x + \sqrt { y } }$$
43. $$\frac { \sqrt { a } - \sqrt { b } } { \sqrt { a } + \sqrt { b } }$$
44. $$\frac { \sqrt { a b } + \sqrt { 2 } } { \sqrt { a b } - \sqrt { 2 } }$$
45. $$\frac { \sqrt { x } } { 5 - 2 \sqrt { x } }$$
46. $$\frac { 1 } { \sqrt { x } - y }$$
47. $$\frac { \sqrt { x } + \sqrt { 2 y } } { \sqrt { 2 x } - \sqrt { y } }$$
48. $$\frac { \sqrt { 3 x } - \sqrt { y } } { \sqrt { x } + \sqrt { 3 y } }$$
49. $$\frac { \sqrt { 2 x + 1 } } { \sqrt { 2 x + 1 } - 1 }$$
50. $$\frac { \sqrt { x + 1 } } { 1 - \sqrt { x + 1 } }$$
51. $$\frac { \sqrt { x + 1 } + \sqrt { x - 1 } } { \sqrt { x + 1 } - \sqrt { x - 1 } }$$
52. $$\frac { \sqrt { 2 x + 3 } - \sqrt { 2 x - 3 } } { \sqrt { 2 x + 3 } + \sqrt { 2 x - 3 } }$$
53. The radius of the base of a right circular cone is given by $$r = \sqrt { \frac { 3 V } { \pi h } }$$ where $$V$$ represents the volume of the cone and $$h$$ represents its height. Find the radius of a right circular cone with volume $$50$$ cubic centimeters and height $$4$$ centimeters. Give the exact answer and the approximate answer rounded to the nearest hundredth.
54. The radius of a sphere is given by $$r = \sqrt [ 3 ] { \frac { 3 V } { 4 \pi } }$$ where $$V$$ represents the volume of the sphere. Find the radius of a sphere with volume $$135$$ square centimeters. Give the exact answer and the approximate answer rounded to the nearest hundredth.

1. $$\frac { \sqrt { 5 } } { 5 }$$

3. $$\frac { \sqrt { 6 } } { 3 }$$

5. $$\frac { \sqrt { 10 } } { 4 }$$

7. $$\frac { 3 - \sqrt { 15 } } { 3 }$$

9. $$\frac { \sqrt { 7 x } } { 7 x }$$

11. $$\frac { \sqrt { a b } } { 5 b }$$

13. $$\frac { \sqrt [ 3 ] { 6 } } { 3 }$$

15. $$\frac { \sqrt [ 3 ] { 2 x ^ { 2 } } } { 2 x }$$

17. $$\frac { 3 \sqrt [ 3 ] { 6 x ^ { 2 } y } } { y }$$

19. $$\frac { \sqrt [ 3 ] { 9 a b } } { 2 b }$$

21. $$\frac { \sqrt [ 5 ] { 9 x ^ { 3 } y ^ { 4 } } } { x y }$$

23. $$\frac { \sqrt [ 5 ] { 27 a ^ { 2 } b ^ { 4 } } } { 3 }$$

25. $$\frac { \sqrt [ 5 ] { 12 x y ^ { 3 } z ^ { 4 } } } { 2 y z }$$

27. $$3\sqrt { 10 } + 9$$

29. $$\frac { \sqrt { 5 } - \sqrt { 3 } } { 2 }$$

31. $$- 1 + \sqrt { 2 }\ 33. \(\frac { - 5 - 3 \sqrt { 5 } } { 2 }$$

35. $$- 4 - \sqrt { 15 }$$

37. $$\frac { 15 - 7 \sqrt { 6 } } { 23 }$$

39. $$\sqrt { x } - \sqrt { y }$$

41. $$\frac { x ^ { 2 } + 2 x \sqrt { y } + y } { x ^ { 2 } - y }$$

43. $$\frac { a - 2 \sqrt { a b + b } } { a - b }$$

45. $$\frac { 5 \sqrt { x } + 2 x } { 25 - 4 x }$$

47. $$\frac { x \sqrt { 2 } + 3 \sqrt { x y } + y \sqrt { 2 } } { 2 x - y }$$

49. $$\frac { 2 x + 1 + \sqrt { 2 x + 1 } } { 2 x }$$

51. $$x + \sqrt { x ^ { 2 } - 1 }$$

53. $$\frac { 5 \sqrt { 6 \pi } } { 2 \pi }$$ centimeters; $$3.45$$ centimeters

Exercise $$\PageIndex{7}$$

1. Research and discuss some of the reasons why it is a common practice to rationalize the denominator.
2. Explain in your own words how to rationalize the denominator.
18The factors $$(a+b)$$ and $$(a-b)$$ are conjugates.