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
Rationalizing a Binomial Denominator Practice Test:

#1:

Instructions: Simplify each.

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


#2:

Instructions: Simplify each.

a) $$\frac{3}{-4 - \sqrt{15}}$$


#3:

Instructions: Simplify each.

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


#4:

Instructions: Simplify each.

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


#5:

Instructions: Simplify each.

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


Written Solutions:

#1:

Solutions:

a) $$\frac{25\sqrt{6} - 15}{47}$$


#2:

Solutions:

a) $$-12 + 3\sqrt{15}$$


#3:

Solutions:

a) $$\frac{5\sqrt{x}}{5x^2 - 6x}$$


#4:

Solutions:

a) $$\frac{-20\sqrt{2n} - 4\sqrt{3} + 10n\sqrt{6} + 6\sqrt{n}}{50n^2 - 3n}$$


#5:

Solutions:

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