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Algebra and Precalculus A Practical Guide

Cameron Geisler

Section 6.1 Multiplying and Dividing Radical Expressions

Rationalizing the denominator involving converting an irrational denominator (a radical that cannot be simplified to a perfect square) to something that doesn’t have any radicals.
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Example 6.1.1. Rationalizing a Simple Fraction.

Consider the fraction \(\frac{1}{\sqrt{2}}\text{,}\) which has \(\sqrt{2}\) in the denominator. To eliminate the square root in the denominator, we can multiply it by \(\sqrt{2}\) (since \(\sqrt{2} \cdot \sqrt{2} = 2\)). If we multiply the denominator by \(\sqrt{2}\text{,}\) we also have to multiply the numerator, so that the number’s value is not changed. Then,
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\begin{align*} \frac{1}{\sqrt{2}} \amp = \frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}}\\ \amp = \frac{\sqrt{2}}{2} \end{align*}
Therefore,
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\begin{gather*} \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \end{gather*}
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Rationalizing the denominator only changes the form of the number, not its value. Most of the time, it is considered “poor form” to leave a radical in the denominator of a final answer (this is analogous to giving a fraction final answer not in reduced form, like putting \(\frac{6}{8}\) instead of \(\frac{3}{4}\)). Basically, it is considered to “look nicer”, so it is a standard form to write final answers, so that it is easy to compare if two answers are the same.
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Subsection 6.1.1 Rationalizing the Denominator

In general, to rationalize the denominator with a square root,
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  1. (Ideally) Simplify radicals, if possible.
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  2. Multiply numerator and denominator by the square root in the denominator.
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Exercise Group 6.1.1. Rationalizing Numerical Monomial Denominators.

Simplify each expression, by rationalizing the denominator.
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(a)
\(\frac{1}{\sqrt{5}}\)
Answer.
\(\frac{\sqrt{5}}{5}\)
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(b)
\(\frac{5}{2\sqrt{3}}\)
Answer.
\(\frac{5\sqrt{3}}{6}\)
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(c)
\(\frac{8}{\sqrt{2}}\)
Answer.
\(4\sqrt{2}\)
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(d)
\(\frac{2}{\sqrt{17}}\)
Answer.
\(\frac{2\sqrt{17}}{17}\)
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(e)
\(\frac{1}{\sqrt{5}}\)
Answer.
\(\frac{\sqrt{5}}{5}\)
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(f)
\(\frac{20}{\sqrt{10}}\)
Answer.
\(2\sqrt{10}\)
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(g)
\(\frac{\sqrt{21}}{\sqrt{24}}\)
Answer.
\(\frac{\sqrt{14}}{4}\)
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(h)
\(-\frac{2}{\sqrt{22}}\)
Answer.
\(-\frac{\sqrt{22}}{11}\)
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(i)
\(\frac{3}{\sqrt{2}}\)
Answer.
\(\frac{3\sqrt{2}}{2}\)
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(j)
\(\frac{\sqrt{3} + 2\sqrt{2}}{2\sqrt{3}}\)
Answer.
\(\frac{3 + 2\sqrt{6}}{6}\)
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(k)
\(\frac{2\sqrt{51}}{\sqrt{3}}\)
Answer.
\(2\sqrt{17}\)
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(l)
\(\frac{\sqrt{5} + \sqrt{2}}{\sqrt{6}}\)
Answer.
\(\frac{\sqrt{30} + 2\sqrt{3}}{6}\)
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(m)
\(\frac{2}{\sqrt{3}}\)
Answer.
\(\frac{2\sqrt{3}}{3}\)
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(n)
\(\frac{1}{5\sqrt{3}}\)
Answer.
\(\frac{\sqrt{3}}{15}\)
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(o)
\(\frac{\sqrt{22} - \sqrt{18}}{\sqrt{10}}\)
Answer.
\(\frac{\sqrt{55}-3\sqrt{5}}{5}\)
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(p)
\(\frac{3\sqrt{500}}{-\sqrt{27}}\)
Answer.
\(-\frac{10\sqrt{15}}{3}\)
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(q)
\(\frac{15\sqrt{18}-3\sqrt{242}}{-3\sqrt{8}}\)
Answer.
\(-2\)
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(r)
\(\frac{3+\sqrt{2}}{\sqrt{2}}\)
Answer.
\(\frac{2+3\sqrt{2}}{2}\)
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(s)
\(\frac{5-\sqrt{2}}{\sqrt{3}}\)
Answer.
\(\frac{5\sqrt{3}-\sqrt{6}}{3}\)
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(t)
\(\frac{2\sqrt{5}+4}{\sqrt{5}}\)
Answer.
\(\frac{10+4\sqrt{5}}{5}\)
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(u)
\(\frac{5\sqrt{8}-2\sqrt{5}}{\sqrt{6}}\)
Answer.
\(\frac{10\sqrt{3}-\sqrt{30}}{3}\)
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(v)
\(\frac{-3\sqrt{12}+2\sqrt{3}}{\sqrt{18}}\)
Answer.
\(\frac{-2\sqrt{6}}{3}\)
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(w)
\(\frac{4\sqrt{2}-6\sqrt{5}}{2\sqrt{3}}\)
Answer.
\(\frac{2\sqrt{6}-3\sqrt{15}}{3}\)
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(x)
\(\frac{4\sqrt{5x}}{3\sqrt{2}}\)
Answer.
\(\frac{2\sqrt{10x}}{3}\)
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Exercise Group 6.1.2. Rationalizing Variable Monomial Denominators.

Simplify each expression, by rationalizing the denominator.
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(a)
\(-\frac{\sqrt{21}}{\sqrt{7x}}\)
Answer.
\(-\frac{\sqrt{3x}}{x}\)
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(b)
\(\frac{18 \sqrt{3x}}{\sqrt{24x}}\)
Answer.
\(\frac{9\sqrt{2}}{2}\)
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(c)
\(\frac{5 - \sqrt{3x^4}}{\sqrt{8x}}\)
Answer.
\(\frac{5\sqrt{2x} - x^2 \sqrt{6x}}{4x}\)
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(d)
\(\frac{3r^2 - \sqrt{5r^2}}{\sqrt{8r}}\)
Answer.
\(\frac{3r\sqrt{2r} - \sqrt{10r}}{4}\)
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(e)
\(\frac{3\sqrt{135 x^5}}{\sqrt{21x^3}}\)
Answer.
\(\frac{9x \sqrt{35}}{7}\)
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(f)
\(\frac{\sqrt{24x^2}}{\sqrt{3x}}\)
Answer.
\(2\sqrt{2x}\)
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(g)
\(\frac{1}{\sqrt{7x}}\)
Answer.
\(\frac{\sqrt{7x}}{7x}\)
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(h)
\(\frac{x}{\sqrt{x^5}}\)
Answer.
\(\frac{\sqrt{x}}{x^2}\)
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(i)
\(\frac{2x}{\sqrt{8x^5}}\)
Answer.
\(\frac{\sqrt{2x}}{2x^2}\)
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(j)
\(\frac{1}{\sqrt{x-2}}\)
Answer.
\(\frac{\sqrt{x-2}}{x-2}\)
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(k)
\(\frac{\sqrt{x}-2}{\sqrt{x}}\)
Answer.
\(\frac{x-2\sqrt{x}}{x}\)
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Subsection 6.1.2 Rationalizing the Denominator with Binomial Denominator

Exercise Group 6.1.3. Rationalizing Binomial Denominators.

Simplify each expression, by rationalizing the denominator.
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(a)
\(\frac{3}{9-\sqrt{5}}\)
Answer.
\(\frac{27+3\sqrt{5}}{76}\)
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(b)
\(\frac{4}{\sqrt{3} + 1}\)
Answer.
\(2\sqrt{3} - 2\)
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(c)
\(\frac{4}{3-\sqrt{2}}\)
Answer.
\(\frac{12+4\sqrt{2}}{7}\)
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(d)
\(\frac{2+\sqrt{3}}{2-\sqrt{3}}\)
Answer.
\(7+4\sqrt{3}\)
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(e)
\(\frac{5\sqrt{3}}{4-\sqrt{6}}\)
Answer.
\(\frac{4\sqrt{3}+3\sqrt{2}}{2}\)
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(f)
\(\frac{-45}{\sqrt{11} + \sqrt{2}}\)
Answer.
\(-5\sqrt{11}+5\sqrt{2}\)
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(g)
\(\frac{5}{\sqrt{3} - 1}\)
Answer.
\(\frac{5\sqrt{3}+5}{2}\)
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(h)
\(\frac{\sqrt{3}+\sqrt{13}}{\sqrt{3}-\sqrt{13}}\)
Answer.
\(\frac{-8-\sqrt{39}}{5}\)
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(i)
\(\frac{5}{9-\sqrt{7}}\)
Answer.
\(\frac{45 + 5\sqrt{7}}{74}\)
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(j)
\(\frac{5\sqrt{2} + \sqrt{3}}{4 \sqrt{2} - \sqrt{3}}\)
Answer.
\(\frac{43+9\sqrt{6}}{29}\)
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(k)
\(\frac{4}{8+\sqrt{5}}\)
Answer.
\(\frac{32-4\sqrt{5}}{59}\)
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(l)
\(\frac{8}{\sqrt{6}+4}\)
Answer.
\(\frac{16-4\sqrt{6}}{5}\)
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(m)
\(\frac{11}{\sqrt{5}+7}\)
Answer.
\(\frac{7-\sqrt{5}}{4}\)
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(n)
\(\frac{2}{3\sqrt{5}-4}\)
Answer.
\(\frac{6\sqrt{5}+8}{29}\)
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(o)
\(\frac{5}{2-\sqrt{3}}\)
Answer.
\(10+5\sqrt{3}\)
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(p)
\(\frac{7\sqrt{2}}{\sqrt{6}+8}\)
Answer.
\(\frac{28\sqrt{2}-7\sqrt{3}}{29}\)
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(q)
\(\frac{2}{1+\sqrt{5}}\)
Answer.
\(\frac{\sqrt{5}-1}{2}\)
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(r)
\(\frac{5}{2+\sqrt{3}}\)
Answer.
\(10-5\sqrt{3}\)
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Exercise Group 6.1.4. Rationalizing Binomial Denominators Continued.

Simplify each expression, by rationalizing the denominator.
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(a)
\(\frac{-\sqrt{7}}{\sqrt{5}-2\sqrt{2}}\)
Answer.
\(\frac{\sqrt{35}+2\sqrt{14}}{3}\)
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(b)
\(\frac{\sqrt{12}}{\sqrt{3}+1}\)
Answer.
\(3-\sqrt{3}\)
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(c)
\(\frac{\sqrt{18}}{\sqrt{2}-1}\)
Answer.
\(6+3\sqrt{2}\)
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(d)
\(\frac{\sqrt{5}}{\sqrt{2}-\sqrt{3}}\)
Answer.
\(-\sqrt{10}-\sqrt{15}\)
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(e)
\(\frac{-\sqrt{5}}{\sqrt{7}-3}\)
Answer.
\(\frac{\sqrt{35}+3\sqrt{5}}{2}\)
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(f)
\(\frac{5\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}}\)
Answer.
\(17+6\sqrt{6}\)
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(g)
\(\frac{6\sqrt{3}-2}{5+4\sqrt{2}}\)
Answer.
\(\frac{-30\sqrt{3}+24\sqrt{6}+10-8\sqrt{2}}{7}\)
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(h)
\(\frac{\sqrt{3} - 5\sqrt{6}}{\sqrt{3} + \sqrt{2}}\)
Answer.
\(3 - 15\sqrt{2} - \sqrt{6} + 10 \sqrt{3}\)
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(i)
\(\frac{3+\sqrt{2}}{1+\sqrt{2}}\)
Answer.
\(2\sqrt{2}-1\)
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(j)
\(\frac{6}{\sqrt{7}-\sqrt{5}}\)
Answer.
\(3\sqrt{7}+3\sqrt{5}\)
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(k)
\(\frac{5\sqrt{8}-3\sqrt{2}}{\sqrt{32}+3\sqrt{2}}\)
Answer.
\(1\)
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(l)
\(\frac{\sqrt{2}-\sqrt{3}}{\sqrt{2}+\sqrt{3}}\)
Answer.
\(2\sqrt{6}-5\)
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(m)
\(\frac{3}{4-\sqrt{5}}\)
Answer.
\(\frac{12+3\sqrt{5}}{11}\)
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(n)
\(\frac{5\sqrt{3x}}{\sqrt{10}+2}\)
Answer.
\(\frac{5\sqrt{30x} - 10 5\sqrt{3x}}{6}\)
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Exercise Group 6.1.5. Rationalizing Complex Binomial Denominators.

Simplify each expression, by rationalizing the denominator.
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(a)
\(\frac{6}{\sqrt{4x}+1}\)
Answer.
\(\frac{12\sqrt{x}-6}{4x-1}\)
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(b)
\(\frac{4x}{\sqrt{6x}+9}\)
Answer.
\(\frac{4x(\sqrt{6x}-9)}{6x-81}\)
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(c)
\(\frac{8}{4-\sqrt{6x}}\)
Answer.
\(\frac{16+4\sqrt{6x}}{8-3x}\)
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(d)
\(\frac{1}{\sqrt{x}-\sqrt{2}}\)
Answer.
\(\frac{\sqrt{x}+\sqrt{2}}{x-2}\)
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(e)
\(\frac{\sqrt{x}-1}{\sqrt{x}+3}\)
Answer.
\(\frac{x-4\sqrt{x}+3}{x-9}\)
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(f)
\(\frac{\sqrt{x}}{\sqrt{x}-\sqrt{y}}\)
Answer.
\(\frac{x+\sqrt{xy}}{x-y}\)
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(g)
\(\frac{\sqrt{y}}{\sqrt{x}+\sqrt{y}}\)
Answer.
\(\frac{\sqrt{xy}-y}{x-y}\)
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(h)
\(\frac{1}{\sqrt{x}-\sqrt{y}}\)
Answer.
\(\frac{\sqrt{x}+\sqrt{y}}{x-y}\)
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(i)
\(\frac{\sqrt{a}+b}{\sqrt{a}-b}\)
Answer.
\(\frac{a+2b\sqrt{a}+b^2}{a-b^2}\)
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(j)
\(\frac{2\sqrt{x}+3\sqrt{y}}{\sqrt{x}-\sqrt{y}}\)
Answer.
\(\frac{2x+5\sqrt{xy}+3y}{x-y}\)
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(k)
\(\frac{3\sqrt{x}}{\sqrt{x} + 3\sqrt{y}}\)
Answer.
\(\frac{3x - 9 \sqrt{xy}}{x - 9y}\)
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(l)
\(\frac{1}{\sqrt{x+1} - 1}\)
Answer.
\(\frac{\sqrt{x+1} + 1}{x}\)
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My recommended calculator Casio FX-991ES Plus C 2nd edition (or some other similar scientific calculators) can rationalize a radical expression (that is numerical, with no variables), both with a monomial denominator, and even with a binomial denominator. Just input the radical expression, and press equals. You can use to at least check your answer.
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Subsection 6.1.3 Rationalizing the Denominator with Higher Roots

In general, multiply the numerator and denominator by whatever is required to get a perfect \(n\)th power in the the radical we want to get rid of. Think: What do you need to make a perfect cube (or 4th power, 5th power, etc.)?
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Exercise Group 6.1.6. Practice with Higher Roots.

Simplify each expression, by rationalizing the denominator.
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(a)
\(\frac{1}{\sqrt[3]{2}}\)
Answer.
\(\frac{\sqrt[3]{4}}{2}\)
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(b)
\(\frac{3}{\sqrt[3]{4}}\)
Answer.
\(\frac{3\sqrt[3]{2}}{2}\)
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(c)
\(\frac{4}{\sqrt[4]{3}}\)
Answer.
\(\frac{4\sqrt[4]{27}}{3}\)
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(d)
\(\frac{4}{\sqrt[3]{2}}\)
Answer.
\(2\sqrt[3]{4}\)
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(e)
\(\frac{1}{\sqrt[5]{x^2}}\)
Answer.
\(\frac{\sqrt[5]{x^3}}{x}\)
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(f)
\(\frac{2}{\sqrt[4]{y}}\)
Answer.
\(\frac{2\sqrt[4]{y^3}}{y}\)
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(g)
\(\frac{4\sqrt{11}}{x\sqrt[3]{6}}\)
Answer.
\(\frac{2\sqrt{11}\sqrt[3]{36}}{3x}\)
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(h)
\(-\frac{7}{2\sqrt[3]{9x}}\)
Answer.
\(\frac{-7\sqrt[3]{3x^2}}{6x}\)
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(i)
\(\sqrt[3]{\frac{x^2}{y}}\)
Answer.
\(\frac{\sqrt[3]{x^2y^2}}{y}\)
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(j)
\(\sqrt[5]{\frac{a^3}{b^2}}\)
Answer.
\(\frac{\sqrt[5]{a^3b^3}}{b}\)
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(k)
\(\frac{3}{\sqrt[5]{24x^2}}\)
Answer.
\(\frac{\sqrt[5]{324x^3}}{2x}\)
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(l)
\(\frac{7}{\sqrt[3]{12x^7}}\)
Answer.
\(\frac{7\sqrt[3]{18x^2}}{6x^3}\)
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(m)
\(\frac{1}{\sqrt{5}}-\frac{1}{\sqrt{3}}\)
Answer.
\(\frac{3\sqrt{5}-5\sqrt{3}}{15}\)
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(n)
\(\frac{\sqrt{2}}{\sqrt{12}}-\frac{5\sqrt{3}}{\sqrt{8}}\)
Answer.
\(\frac{-13\sqrt{6}}{12}\)
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(o)
\(\frac{\sqrt{6}}{2\sqrt{5}+3\sqrt{3}}-\frac{\sqrt{2}}{\sqrt{7}-2\sqrt{3}}\)
Answer.
\(\frac{-10\sqrt{30}+45\sqrt{2}+7\sqrt{14}+14\sqrt{6}}{35}\)
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(p)
\(\frac{6 \sqrt[3]{4x^7}}{\sqrt[3]{14x}}\)
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Subsection 6.1.4 Rationalizing the Denominator Historical Note

Rationalizing the denominator of a fraction was more useful before computers, so division had to be done by hand, using long division. Dividing 1 by \(\sqrt{2}\) meant dividing 1 by 1.41421\(\dots\text{.}\) Instead, it is easier to divide \(\sqrt{2}\) by \(2\text{.}\)
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Subsection 6.1.5 Word Problems

Checkpoint 6.1.2. Rectangle Area.

A rectangle has dimensions \(6+\sqrt{3}\) and \(6-\sqrt{3}\text{.}\) Find an expression for its area.
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Answer.
33
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Checkpoint 6.1.3. Reciprocal Proof.

Show that the reciprocal of \(\sqrt{26}-5\) is 10 greater than \(\sqrt{26}-5\text{.}\)
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Hint.
take the reciprocal and rationalize the denominator.
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