Rationalizing Denominators

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Show your work for each problem.

1.
What should you multiply $\frac{3}{\sqrt{5}}$ by to rationalize the denominator?
1. a) $\frac{\sqrt{3}}{\sqrt{3}}$
2. b) $\frac{\sqrt{5}}{\sqrt{5}}$
3. c) $\frac{5}{5}$
4. d) $\frac{3}{3}$
2.
Rationalize $\frac{1}{\sqrt{2}}$ and write the result as $\frac{\sqrt{2}}{\square}$ . What number goes in the box?
3.
What is the conjugate of $(4 + \sqrt{3})$ ?
1. a) $(4 - \sqrt{3})$
2. b) $(4 + \sqrt{3})$
3. c) $(-4 - \sqrt{3})$
4. d) $(-4 + \sqrt{3})$
4.
Rationalize: $\frac{4}{\sqrt{2}}$
5.
Rationalize $\frac{6}{\sqrt{3}}$ , simplify, and write the result as $\square\sqrt{3}$ . What number goes in the box?
6.
What is $(2 + \sqrt{5})(2 - \sqrt{5})$ ?
1. a) $1$
2. b) $-1$
3. c) $9$
4. d) $4 - \sqrt{5}$
7.
Rationalize: $\frac{3}{\sqrt{12}}$
8.
Rationalize $\frac{2}{1 + \sqrt{3}}$ and write the result as $a + b\sqrt{3}$ . What is $a$ ?
9.
What is the rationalized form of $\frac{1}{\sqrt{3}}$ ?
1. a) $\frac{1}{3}$
2. b) $\sqrt{3}$
3. c) $\frac{\sqrt{3}}{3}$
4. d) $\frac{3}{\sqrt{3}}$
10.
Rationalize: $\frac{5}{2 - \sqrt{3}}$
11.
Rationalize $\frac{6}{\sqrt{5} + \sqrt{2}}$ and write the result as $a\sqrt{5} + b\sqrt{2}$ . What is $b$ ?
12.
Rationalize: $\frac{3}{\sqrt{6} - \sqrt{3}}$
13.
What is $(\sqrt{7} + \sqrt{2})(\sqrt{7} - \sqrt{2})$ ?
1. a) $9$
2. b) $\sqrt{5}$
3. c) $5$
4. d) $\sqrt{14}$
14.
Rationalize: $\frac{4}{3 + \sqrt{7}}$
15.
Rationalize $\frac{\sqrt{3}}{\sqrt{5} - \sqrt{2}}$ and write the result as $\frac{\sqrt{a} + \sqrt{b}}{3}$ with $a > b$ . What is $a$ ?
16.
Which expression is in rationalized form?
1. a) $\frac{3}{\sqrt{5}}$
2. b) $\frac{1}{\sqrt{3}}$
3. c) $\frac{2}{1 + \sqrt{2}}$
4. d) $\frac{\sqrt{5}}{3}$