### About Further Operations with Radicals:

Once we have a good understanding of how to simplify a radical, we move into operations with radicals. These operations combine everything that we have learned so far to produce a simplified answer. We will also look at how to rationalize a binomial denominator using a conjugate.

Test Objectives

- Demonstrate the ability to simplify a square root, cube root, or higher-level root
- Demonstrate the ability to perform operations with radicals
- Demonstrate the ability to rationalize a binomial denominator

#1:

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

a) $$\sqrt{3}(4 + 4\sqrt{2})$$

b) $$-3\sqrt{5}(\sqrt{10}+ 5)$$

c) $$\sqrt{6}(\sqrt{10}+ \sqrt{6})$$

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#2:

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

a) $$(7\sqrt{2}+ 2)(-2\sqrt{2}- 5)$$

b) $$(-4\sqrt{5m}+ 5\sqrt{6})(2\sqrt{3m}+ 3\sqrt{6})$$

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#3:

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

a) $$(4\sqrt{3}+ 2)(4\sqrt{3}- 2)$$

b) $$(3\sqrt{2}+ 7)(3\sqrt{2}+ 7)$$

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#4:

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

a) $$\frac{4}{2 - \sqrt{7}}$$

b) $$\frac{\sqrt{5}+ \sqrt{2}}{\sqrt{3}- \sqrt{11}}$$

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#5:

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

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

b) $$\frac{5x - 5\sqrt{5x^2}}{4x + \sqrt{2x^4}}$$

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Written Solutions:

#1:

Solutions:

a) $$4\sqrt{3}+ 4\sqrt{6}$$

b) $$-15\sqrt{2}- 15\sqrt{5}$$

c) $$2\sqrt{15}+ 6$$

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#2:

Solutions:

a) $$-38 - 39\sqrt{2}$$

b) $$-8m\sqrt{15}- 12\sqrt{30m}+ 30\sqrt{2m}+ 90$$

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#3:

Solutions:

a) $$44$$

b) $$42\sqrt{2}+ 67$$

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#4:

Solutions:

a) $$\frac{8 + 4\sqrt{7}}{-3}$$

b) $$\frac{\sqrt{15}+ \sqrt{55}+ \sqrt{6}+ \sqrt{22}}{-8}$$

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#5:

Solutions:

a) $$\frac{4n + 8n\sqrt{5n}+ 5n\sqrt{3}+ 10n\sqrt{15n}}{2 - 40n}$$

b) $$\frac{20 - 5x\sqrt{2}- 20\sqrt{5}+ 5x\sqrt{10}}{16 - 2x^2}$$