About Rationalize 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
Rationalize the Denominator Practice Test:

#1:

Instructions: Simplify each.

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

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

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


#2:

Instructions: Simplify each.

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

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

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


#3:

Instructions: Simplify each.

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

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

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


#4:

Instructions: Simplify each.

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

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


#5:

Instructions: Simplify each.

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

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


Written Solutions:

#1:

Solutions:

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

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

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


#2:

Solutions:

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

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

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


#3:

Solutions:

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

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

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


#4:

Solutions:

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

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


#5:

Solutions:

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

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