### About Rationalizing the Denominator:

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

#1:

Instructions: Simplify each.

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

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

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#2:

Instructions: Simplify each, assume all variables represent positive real numbers.

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

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

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#3:

Instructions: Simplify each, assume all variables represent positive real numbers.

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

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

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#4:

Instructions: Simplify each, assume all variables represent positive real numbers.

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

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#5:

Instructions: Simplify each, assume all variables represent positive real numbers.

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

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Written Solutions:

#1:

Solutions:

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

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

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#2:

Solutions:

a) $$\frac{2\sqrt[3]{9}}{15}$$

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

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#3:

Solutions:

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

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

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#4:

Solutions:

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

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#5:

Solutions:

a) $$\frac{-5\sqrt[4]{108p^3}+ 6\sqrt{2p}}{24}$$