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