### 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, assume all variables represent positive real numbers.

a) $$\frac{5}{5\sqrt{x^3}- 6\sqrt{x}}$$

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#4:

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

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

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#5:

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

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) $$\frac{15a\sqrt{5a}+ 5a\sqrt{3a}+ 15\sqrt{10a}+ 5\sqrt{6a}}{42}$$