A simplified radical expression does not contain any radicals in the denominator. The process we use to clear a denominator of its radical is known as rationalizing the denominator. We rationalize the denominator by multiplying the numerator and denominator by a radical that will transform the denominator into a rational number.

Test Objectives
• Demonstrate the ability to multiply radicals
• Demonstrate the ability to rationalize a denominator with a square root
• Demonstrate the ability to rationalize a denominator with a higher level root
Rationalizing the Denominator Practice Test:

#1:

Instructions: Simplify each.

a) $$\frac{11\sqrt{10}}{8\sqrt{7}}$$

b) $$\frac{2\sqrt{24}}{2\sqrt{56}}$$

#2:

Instructions: Simplify each.

a) $$\frac{2\sqrt{3}}{5\sqrt{9}}$$

b) $$\frac{3\sqrt{5x^4y}}{4\sqrt{2x^3y^3}}$$

#3:

Instructions: Simplify each.

a) $$\frac{5x^2 - 2\sqrt{x^4}}{5\sqrt{12x^3}}$$

b) $$\frac{5\sqrt{3p}- 5\sqrt{2p^2}}{3\sqrt{10p}}$$

#4:

Instructions: Simplify each.

a) $$\frac{3x + 3\sqrt{2x^3}}{3\sqrt{4x^2}}$$

#5:

Instructions: Simplify each.

a) $$\frac{-5p + 2\sqrt{3p^3}}{4\sqrt{12p}}$$

Written Solutions:

#1:

Solutions:

a) $$\frac{11\sqrt{70}}{56}$$

b) $$\frac{\sqrt{21}}{7}$$

#2:

Solutions:

a) $$\frac{2\sqrt{9}}{15}$$

b) $$\frac{3\sqrt{20xy}}{8y}$$

#3:

Solutions:

a) $$\frac{\sqrt{3x}}{{10}}$$

b) $$\frac{\sqrt{30}- 2\sqrt{5p}}{6}$$

#4:

Solutions:

a) $$\frac{\sqrt{2x}+ \sqrt{4x}}{2}$$

#5:

Solutions:

a) $$\frac{-5\sqrt{108p^3}+ 6\sqrt{2p}}{24}$$