6.1.1 and Rationalizing the Denominator

We rationalize the denominator to ensure that it becomes easier to perform any calculation on the rational number. When we rationalize the denominator in a fraction, then we are eliminating any radical expressions such as square roots and cube roots from the denominator. In this article, let's learn about rationalizing the denominator, its meaning, and methods with some examples.

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​Rationalizing the Denominator

Multiple Choice

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

• $$\frac{\sqrt{15}}{6}$$  

• $$\frac{\sqrt{15}}{18}$$  

• $$\frac{5\sqrt{3}}{6}$$  

• $$\frac{5\sqrt{3}}{18}$$  

Multiple Choice

Rationalize the denominator:
$$\frac{2\sqrt{8}}{\sqrt{12}}$$  

• $$\frac{\sqrt{96}}{6}$$  

• $$\frac{2\sqrt{6}}{3}$$  

• $$\frac{2\sqrt{96}}{12}$$  

• $$\frac{4\sqrt{2}}{2\sqrt{3}}$$  

Multiple Choice

Multiply:
$$\left(4+\sqrt{2}\right)\left(4-\sqrt{2}\right)$$  

• $$16-\sqrt{2}$$  

• $$4-\sqrt{2}$$  

• 14

• 18

Multiple Choice

What should you multiply $$\frac{6}{1-\sqrt{5}}$$  by to rationalize the denominator?

• -4

• $$1-\sqrt{5}$$  

• $$1+\sqrt{5}$$  

• $$-\frac{\left(3-3\sqrt{5}\right)}{2}$$  

Multiple Choice

Rationalize the denominator:
$$\frac{7\sqrt{3}}{4\sqrt{7}}$$  

• $$\frac{7\sqrt{21}}{28}$$  

• $$\frac{7\sqrt{21}}{196}$$  

• $$\frac{\sqrt{21}}{28}$$  

• $$\frac{\sqrt{21}}{4}$$  

Multiple Choice

Rationalize the denominator:
$$\frac{\sqrt{4}}{\sqrt{121}}$$  

• $$\frac{4}{11}$$  

• $$\frac{\sqrt{484}}{121}$$  

• $$\frac{2}{11}$$  

• $$\frac{2\sqrt{121}}{121}$$  

Multiple Select

Choose the correct values for x and y in the right triangle.

• $$x=\frac{5\sqrt{6}}{2}$$

• $$x=5$$

• $$y\ =\ 10$$

• $$y=10\sqrt{3}$$

Multiple Choice

What type of special triangle is this?

• 45°-45°-90°
• 30°-60°-90°
• Equiangular
• Equilateral

Multiple Choice

What type of special triangle is this?

• 45°-45°-90°
• 30°-60°-90°
• Isosceles
• Obtuse

Multiple Choice

In this 45-45-90 triangle, I have been given a leg, so to find the other leg I...

• Multiply that leg by 2
• Use the same length for the second leg
• Multiply that leg by √2
• Divide that leg by √2

Multiple Choice

I have been given the short leg in this 30-60-90 triangle.  How do I find the length of the hypotenuse?

• Multiply 4 by 2
• Multiply 4 by √3
• Multiply 4 by √2
• Divide 4 by √3

Multiple Select

Choose the correct values for x and y in the right triangle.

• $$x=12$$

• $$x=12\sqrt{2}$$

• $$y\ =\ 24$$

• $$y=12\sqrt{2}$$

Multiple Select

Choose the correct values for x and y in the right triangle.

• $$x=18$$

• $$x=9\sqrt{2}$$

• $$y\ =\ 9$$

• $$y=9\sqrt{2}$$

Multiple Select

Choose the correct values for x and y in the right triangle.

• $$x=14$$

• $$x=7\sqrt{2}$$

• $$y\ =\ 7$$

• $$y=7\sqrt{2}$$

Multiple Select

Choose the correct values for x and y in the right triangle.

• $$x=3$$

• $$x=3\sqrt{2}$$

• $$y\ =\ 6$$

• $$y=6\sqrt{2}$$

Multiple Select

Choose the correct values for x and y in the right triangle.

• $$x=4\sqrt{3}$$

• $$x=4$$

• $$y\ =\ 8\sqrt{3}$$

• $$y=4\sqrt{6}$$

Multiple Select

Choose the correct values for x and y in the right triangle.

• $$x=3\sqrt{3}$$

• $$x=6$$

• $$y\ =\ 3\sqrt{3}$$

• $$y=6$$

Multiple Select

Choose the correct values for x and y in the right triangle.

• $$x=\frac{11\sqrt{3}}{3}$$

• $$x=11$$

• $$y\ =\ 22$$

• $$y=11\sqrt{3}$$