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Rationalizing the Denominator Test
About Rationalizing the Denominator:

When working with radicals, we always want to report a simplified result. One rule of simplification states that we can’t have a radical in the denominator. In order to deal with this problem, we follow a procedure known as rationalizing the denominator. This will give us a rational number in the denominator.

Test Objectives:

•Demonstrate the ability to simplify a square root, cube root, or higher level root

•Demonstrate the ability to rationalize a denominator that contains a square root

•Demonstrate the ability to rationalize a denominator that contains a cube root, or higher level root

Rationalizing the Denominator Test:




#1:


Instructions: Simplify each.


a) $$\frac{\sqrt{4}}{\sqrt{6}}$$


b) $$\frac{\sqrt{8}}{4\sqrt{6}}$$


c) $$\frac{\sqrt{5}}{6\sqrt{8}}$$


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#2:


Instructions: Simplify each.


a) $$\frac{3\sqrt{5}}{5\sqrt{7}}$$


b) $$\frac{\sqrt{2}}{\sqrt{7}}$$


c) $$\frac{\sqrt{49}}{3\sqrt{14}}$$


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#3:


Instructions: Simplify each.


a) $$\frac{\sqrt{32}}{\sqrt{20}}$$


b) $$\frac{\sqrt{3}}{4\sqrt{5}}$$


c) $$\frac{\sqrt{10}}{\sqrt{15}}$$


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#4:


Instructions: Simplify each.


a) $$\frac{8\sqrt{10} + \sqrt{5}}{13}$$


b) $$\frac{5 + 4\sqrt{3}}{10\sqrt{31}}$$


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#5:


Instructions: Simplify each.


a) $$\frac{3}{\sqrt[3]{3x}}$$


b) $$\frac{u}{\sqrt[4]{216}}$$


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Written Solutions:




#1:


Solution:


a) $$\frac{\sqrt{6}}{3}$$


b) $$\frac{\sqrt{3}}{6}$$


c) $$\frac{\sqrt{10}}{24}$$


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#2:


Solution:


a) $$\frac{3\sqrt{35}}{35}$$


b) $$\frac{\sqrt{14}}{7}$$


c) $$\frac{\sqrt{14}}{6}$$


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#3:


Solution:


a) $$\frac{2\sqrt{10}}{5}$$


b) $$\frac{\sqrt{15}}{20}$$


c) $$\frac{\sqrt{6}}{3}$$


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#4:


Solution:


a) $$\frac{8\sqrt{130} + \sqrt{65}}{13}$$


b) $$\frac{5\sqrt{31} + 4\sqrt{93}}{310}$$


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#5:


Solution:


a) $$\frac{\sqrt[3]{9x^2}}{x}$$


b) $$\frac{\sqrt[4]{6u^3}}{6}$$


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