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Further Operations with Radicals Test
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 a binomial denominator

Further Operations with Radicals Test:




#1:


Instructions: Simplify each.


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.


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.


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.


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


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


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#5:


Instructions: Simplify each.


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:


Solution:


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


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


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


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#2:


Solution:


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


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


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#3:


Solution:


a) $$44$$


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


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#4:


Solution:


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


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


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#5:


Solution:


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


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


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