About Rationalizing a Binomial Denominator:
A simplified radical expression does not contain any radicals in the denominator. In some cases, we will face a two-term denominator that contains radicals. For this scenario, we can’t use the same methods from rationalizing with a single-term radical in the denominator. To rationalize a binomial denominator, we multiply numerator and denominator by the conjugate of the denominator.
Test Objectives
- Demonstrate the ability to multiply and simplify radicals
- Demonstrate the ability to find the conjugate of the denominator
- Demonstrate the ability to rationalize a binomial denominator
#1:
Instructions: Simplify each.
a) $$\frac{15}{5\sqrt{6}+ 3}$$
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#2:
Instructions: Simplify each.
a) $$\frac{3}{-4 - \sqrt{15}}$$
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#3:
Instructions: Simplify each.
a) $$\frac{5}{5\sqrt{x^3}- 6\sqrt{x}}$$
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#4:
Instructions: Simplify each.
a) $$\frac{-4 + 2\sqrt{3n}}{5\sqrt{2n^3}- \sqrt{3n^2}}$$
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#5:
Instructions: Simplify each.
a) $$\frac{5a^3 + 5\sqrt{2a^4}}{3\sqrt{5a^3}- \sqrt{3a^3}}$$
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Written Solutions:
#1:
Solutions:
a) $$\frac{25\sqrt{6}- 15}{47}$$
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#2:
Solutions:
a) $$-12 + 3\sqrt{15}$$
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#3:
Solutions:
a) $$\frac{5\sqrt{x}}{5x^2 - 6x}$$
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#4:
Solutions:
a) $$\frac{-20\sqrt{2n}- 4\sqrt{3}+ 10n\sqrt{6}+ 6\sqrt{n}}{50n^2 - 3n}$$
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#5:
Solutions:
a) $$\frac{15a\sqrt{5a}+ 5a\sqrt{3a}+ 15\sqrt{10a}+ 5\sqrt{6a}}{42}$$
