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# Multiplication and Division of Radicals

## Rationalize the denominator

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Practice Multiplication and Division of Radicals
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Dividing Square Roots

The area of a rectangle is $\sqrt{30}$ . The length of the rectangle is $\sqrt{20}$ . What is the width of the rectangle?

### Watch This

Watch the first part of this video, until about 3:15.

### Guidance

Dividing radicals can be a bit more difficult that the other operations. The main complication is that you cannot leave any radicals in the denominator of a fraction. For this reason we have to do something called rationalizing the denominator , where you multiply the top and bottom of a fraction by the same radical that is in the denominator. This will cancel out the radicals and leave a whole number.

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

5. $\frac{\sqrt{a}}{\sqrt{b}} \cdot \frac{\sqrt{b}}{\sqrt{b}} = \frac{\sqrt{ab}}{b}$

#### Example A

Simplify $\sqrt{\frac{1}{4}}$ .

Solution: Break apart the radical by using Rule #4.

$\sqrt{\frac{1}{4}}=\frac{\sqrt{1}}{\sqrt{4}}=\frac{1}{2}$

#### Example B

Simplify $\frac{2}{\sqrt{3}}$ .

Solution: This might look simplified, but radicals cannot be in the denominator of a fraction. This means we need to apply Rule #5 to get rid of the radical in the denominator, or rationalize the denominator. Multiply the top and bottom of the fraction by $\sqrt{3}$ .

$\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{3}}{3}$

#### Example C

Simplify $\sqrt{\frac{32}{40}}$ .

Solution: Reduce the fraction, and then apply the rules above.

$\sqrt{\frac{32}{40}}= \sqrt{\frac{4}{5}}= \frac{\sqrt{4}}{\sqrt{5}}= \frac{2}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}=\frac{2\sqrt{5}}{5}$

Intro Problem Revisit Recall that the area of a rectangle equals the length times the width, so to find the width, we must divide the area by the length.

$\sqrt{\frac{30}{20}}$ = $\sqrt{\frac{3}{2}}$ .

Now we need to rationalize the denominator. Multiply the top and bottom of the fraction by $\sqrt{2}$ .

$\frac{\sqrt{3}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{6}}{2}$

Therefore, the width of the rectangle is $\frac{\sqrt{6}}{2}$ .

### Guided Practice

Simplify the following expressions using the Radical Rules learned in this concept and the previous concept.

1. $\sqrt{\frac{1}{2}}$

2. $\sqrt{\frac{64}{50}}$

3. $\frac{4\sqrt{3}}{\sqrt{6}}$

1. $\sqrt{\frac{1}{2}}=\frac{\sqrt{1}}{\sqrt{2}}=\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}= \frac{\sqrt{2}}{2}$

2. $\sqrt{\frac{64}{50}}=\sqrt{\frac{32}{25}}=\frac{\sqrt{16 \cdot 2}}{5}= \frac{4\sqrt{2}}{5}$

3. The only thing we can do is rationalize the denominator by multiplying the numerator and denominator by $\sqrt{6}$ and then simplify the fraction.

$\frac{4\sqrt{3}}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}}= \frac{4\sqrt{18}}{6}=\frac{4\sqrt{9 \cdot 2}}{6}= \frac{12\sqrt{2}}{6}=2\sqrt{2}$

### Vocabulary

Rationalize the denominator
The process used to get a radical out of the denominator of a fraction.

### Practice

Simplify the following fractions.

1. $\sqrt{\frac{4}{25}}$
2. $-\sqrt{\frac{16}{49}}$
3. $\sqrt{\frac{96}{121}}$
4. $\frac{5\sqrt{2}}{\sqrt{10}}$
5. $\frac{6}{\sqrt{15}}$
6. $\sqrt{\frac{60}{35}}$
7. $8\frac{\sqrt{18}}{\sqrt{30}}$
8. $\frac{12}{\sqrt{6}}$
9. $\sqrt{\frac{208}{143}}$
10. $\frac{21\sqrt{3}}{2\sqrt{14}}$

Challenge Use all the Radical Rules you have learned in the last two oncepts to simplify the expressions.

1. $\sqrt{\frac{8}{12}} \cdot \sqrt{15}$
2. $\sqrt{\frac{32}{45}} \cdot \frac{6\sqrt{20}}{\sqrt{5}}$
3. $\frac{\sqrt{24}}{\sqrt{2}}+\frac{8\sqrt{26}}{\sqrt{8}}$
4. $\frac{\sqrt{2}}{\sqrt{3}}+\frac{4\sqrt{6}}{\sqrt{3}}$
5. $\frac{5\sqrt{5}}{\sqrt{12}}-\frac{2\sqrt{15}}{\sqrt{10}}$

### Vocabulary Language: English

Rationalize the denominator

Rationalize the denominator

To rationalize the denominator means to rewrite the fraction so that the denominator no longer contains a radical.