Rationalizing Denominators

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) $\frac{3}{3}$
Answer: $\frac{\sqrt{5}}{\sqrt{5}}$
Multiply by $\frac{\sqrt{5}}{\sqrt{5}} = 1$ . This gives $\sqrt{5} \cdot \sqrt{5} = 5$ in the denominator, eliminating the radical.
2.
Rationalize $\frac{1}{\sqrt{2}}$ and write the result as $\frac{\sqrt{2}}{\square}$ . What number goes in the box?
Answer: 2
$\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}$ . This is the exact value of $\sin(45°)$ !
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) $(-4 + \sqrt{3})$
Answer: $(4 - \sqrt{3})$
The conjugate of $(a + b)$ is $(a - b)$ . So the conjugate of $(4 + \sqrt{3})$ is $(4 - \sqrt{3})$ .
4.
Rationalize: $\frac{4}{\sqrt{2}}$
Answer: $2\sqrt{2}$
1. What do you multiply the numerator and the denominator by? sqrt(2)
2. What is $4 \times \sqrt{2}$ ? 4*sqrt(2)
3. What is $\sqrt{2} \times \sqrt{2}$ ? 2
4. $\frac{4\sqrt{2}}{2} = \square\sqrt{2}$ . What number goes in the box? 2
5.
Rationalize $\frac{6}{\sqrt{3}}$ , simplify, and write the result as $\square\sqrt{3}$ . What number goes in the box?
Answer: 2
$\frac{6}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{6\sqrt{3}}{3} = 2\sqrt{3}$
6.
What is $(2 + \sqrt{5})(2 - \sqrt{5})$ ?
1. a) $1$
2. b) $-1$
3. c) $9$
4. d) $4 - \sqrt{5}$
Answer: $-1$
$(2 + \sqrt{5})(2 - \sqrt{5}) = 2^2 - (\sqrt{5})^2 = 4 - 5 = -1$
7.
Rationalize: $\frac{3}{\sqrt{12}}$
Answer: $\frac{\sqrt{3}}{2}$
1. Simplify $\sqrt{12}$ first: $\sqrt{12} = \square\sqrt{3}$ . What number goes in the box? 2
2. The fraction is now $\frac{3}{2\sqrt{3}}$ . What do you multiply the numerator and the denominator by? sqrt(3)
3. What is the new denominator $2\sqrt{3} \cdot \sqrt{3}$ ? 6
4. $\frac{3\sqrt{3}}{6} = \frac{\sqrt{3}}{\square}$ . What number goes in the box? 2
8.
Rationalize $\frac{2}{1 + \sqrt{3}}$ and write the result as $a + b\sqrt{3}$ . What is $a$ ?
Answer: -1
$\frac{2}{1 + \sqrt{3}} \cdot \frac{1 - \sqrt{3}}{1 - \sqrt{3}} = \frac{2(1 - \sqrt{3})}{1 - 3} = \frac{2 - 2\sqrt{3}}{-2} = -1 + \sqrt{3}$ So $a = -1$ .
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) $\frac{3}{\sqrt{3}}$
Answer: $\frac{\sqrt{3}}{3}$
$\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{3}$ . This equals $\tan(30°)$ .
10.
Rationalize: $\frac{5}{2 - \sqrt{3}}$
Answer: $10 + 5\sqrt{3}$
1. What is the conjugate of $(2 - \sqrt{3})$ ? 2 + sqrt(3)
2. Multiply the numerator: $5(2 + \sqrt{3})$ 10 + 5*sqrt(3)
3. Multiply the denominators: $(2 - \sqrt{3})(2 + \sqrt{3})$ 1
4. Divide by the denominator: $\frac{10 + 5\sqrt{3}}{1} = 10 + \square\sqrt{3}$ . What number goes in the box? 5
11.
Rationalize $\frac{6}{\sqrt{5} + \sqrt{2}}$ and write the result as $a\sqrt{5} + b\sqrt{2}$ . What is $b$ ?
Answer: -2
$\frac{6}{\sqrt{5} + \sqrt{2}} \cdot \frac{\sqrt{5} - \sqrt{2}}{\sqrt{5} - \sqrt{2}} = \frac{6(\sqrt{5} - \sqrt{2})}{5 - 2} = \frac{6\sqrt{5} - 6\sqrt{2}}{3} = 2\sqrt{5} - 2\sqrt{2}$ So $b = -2$ .
12.
Rationalize: $\frac{3}{\sqrt{6} - \sqrt{3}}$
Answer: $\sqrt{6} + \sqrt{3}$
1. What is the conjugate? sqrt(6) + sqrt(3)
2. Multiply numerator: $3(\sqrt{6} + \sqrt{3})$ 3*sqrt(6) + 3*sqrt(3)
3. Apply difference of squares to denominator: $(\sqrt{6})^2 - (\sqrt{3})^2$ 3
4. Divide each term by 3: $\frac{3\sqrt{6} + 3\sqrt{3}}{3}$ . What number is now in front of each radical? 1
13.
What is $(\sqrt{7} + \sqrt{2})(\sqrt{7} - \sqrt{2})$ ?
1. a) $9$
2. b) $\sqrt{5}$
3. c) $5$
4. d) $\sqrt{14}$
Answer: $5$
$(\sqrt{7})^2 - (\sqrt{2})^2 = 7 - 2 = 5$ . The product of two conjugates has no radical left.
14.
Rationalize: $\frac{4}{3 + \sqrt{7}}$
Answer: $6 - 2\sqrt{7}$
1. What is the conjugate of $(3 + \sqrt{7})$ ? 3 - sqrt(7)
2. Multiply the numerator: $4(3 - \sqrt{7})$ 12 - 4*sqrt(7)
3. Apply difference of squares to denominator: $(3)^2 - (\sqrt{7})^2$ 2
4. $\frac{12 - 4\sqrt{7}}{2} = 6 - \square\sqrt{7}$ . What number goes in the box? 2
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 $a$ ?
Answer: 15
$\frac{\sqrt{3}}{\sqrt{5} - \sqrt{2}} \cdot \frac{\sqrt{5} + \sqrt{2}}{\sqrt{5} + \sqrt{2}} = \frac{\sqrt{3}(\sqrt{5} + \sqrt{2})}{5 - 2} = \frac{\sqrt{15} + \sqrt{6}}{3}$ So $a = 15$ .
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) $\frac{\sqrt{5}}{3}$
Answer: $\frac{\sqrt{5}}{3}$
Only $\frac{\sqrt{5}}{3}$ has a rational denominator (3). The others all have radicals in their denominators.