### About Simplifying Radicals:

Once we understand how to evaluate a square root, cube root, and higher-level roots, it’s time to learn how to perform operations with roots. We will learn how to multiply and divide roots. In the process, we will learn how to simplify roots. This is very important, as we always want to report a simplified answer.

Test Objectives

- Demonstrate the ability to simplify a square root, cube root, or higher-level root
- Demonstrate the ability to find the product of two or more roots
- Demonstrate the ability to perform a division with roots

#1:

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

a) $$\sqrt{125}$$

b) $$\sqrt{8}$$

c) $$\sqrt{18}$$

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#2:

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

a) $$\sqrt{10}\cdot \sqrt{3}$$

b) $$\sqrt{6}\cdot \sqrt{12}$$

c) $$4\sqrt{5r^2}\cdot 4\sqrt{3r^2}$$

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#3:

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

a) $$4\sqrt{15x^2}\cdot 5\sqrt{6x^3}$$

b) $$\frac{\sqrt{12}}{4\sqrt{3}}$$

c) $$\frac{\sqrt{8}}{\sqrt{16}}$$

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#4:

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

a) $$\frac{3\sqrt{2}}{2\sqrt{18}}$$

b) $$\frac{3\sqrt{6}}{2\sqrt{8}}$$

c) $$2\sqrt[3]{750xy^4}$$

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#5:

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

a) $$2\sqrt[4]{162x^8y^5}$$

b) $$7\sqrt[4]{405x^6y^5}$$

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Written Solutions:

#1:

Solutions:

a) $$5\sqrt{5}$$

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

c) $$3\sqrt{2}$$

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#2:

Solutions:

a) $$\sqrt{30}$$

b) $$6\sqrt{2}$$

c) $$16r^2\sqrt{15}$$

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#3:

Solutions:

a) $$60x^2\sqrt{10x}$$

b) $$\frac{1}{2}$$

c) $$\frac{\sqrt{2}}{2}$$

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#4:

Solutions:

a) $$\frac{1}{2}$$

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

c) $$10y\sqrt[3]{6xy}$$

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#5:

Solutions:

a) $$6x^2y\sqrt[4]{2y}$$

b) $$21xy\sqrt[4]{5x^2y}$$