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Wize High School Grade 12 Calculus Textbook > Review

Radical Expressions

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Radical Expressions

We can convert radicals to exponents: (amn)=(an)m=(amn)\displaystyle \left(\sqrt[n]{a^m}\right)=\left(\sqrt[n]{a}\right)^m=\left(a^{\frac{m}{n}}\right)
*Special cases:
  • aa=a\sqrt a \sqrt a=a
  • (a+b)(a−b)=a−b\left(\sqrt{a}+\sqrt{b}\right)\left(\sqrt{a}-\sqrt{b}\right)=a-b
  • (a+b)(a−b)=a2−b\left(a+\sqrt{b}\right)\left(a-\sqrt{b}\right)=a^2-b

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Rationalize the Denominator

When we have an expression with a radical in the denominator, we can rationalize the denominator:

Case 1: Single radical term in the denominator
ab=ab×bb=abb\displaystyle \boxed{\frac{a}{\sqrt{b}}}=\frac{a}{\sqrt{b}}\times\frac{\sqrt{b}}{\sqrt{b}}=\frac{a\sqrt{b}}{b}
  • We multiply the numerator and denominator by the radical in the denominator
  • Example: 96=\displaystyle \frac{9}{\sqrt{6}}=
96×66=966=362 \displaystyle \frac{9}{\sqrt{6}}\times\frac{\sqrt{6}}{\sqrt{6}}=\frac{9\sqrt{6}}{6}=\frac{3\sqrt{6}}{2\ }


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Case 2: More than a single radical term in the denominator
ab−c=ab−c×b+cb+c=a(b+c)b−c\displaystyle \boxed{\frac{a}{\sqrt{b}-\sqrt{c}}}=\frac{a}{\sqrt{b}-\sqrt{c}}\times\frac{\sqrt{b}+\sqrt{c}}{\sqrt{b}+\sqrt{c}}=\frac{a\left(\sqrt{b}+\sqrt{c}\right)}{b-c}
  • We multiply the numerator and denominator by the conjugate of the radical that is in the denominator
  • Example: 35−6=\displaystyle \frac{3}{\sqrt{5}-\sqrt{6}}=
35−6×5+65+6=3(5+6)5−6=−3(5+6)\displaystyle \frac{3}{\sqrt{5}-\sqrt{6}}\times\frac{\sqrt{5}+\sqrt{6}}{\sqrt{5}+\sqrt{6}}=\frac{3\left(\sqrt{5}+\sqrt{6}\right)}{5-6}=-3\left(\sqrt{5}+\sqrt{6}\right)

  • Example: 13+5=\displaystyle \frac{1}{\sqrt{3}+\sqrt{5}}=
13+5 ×3−53−5=3−53−5=3−5−2=5−32\displaystyle \frac{1}{\sqrt{3}+\sqrt{5}\ }\times\frac{\sqrt{3}-\sqrt{5}}{\sqrt{3}-\sqrt{5}}=\frac{\sqrt{3}-\sqrt{5}}{3-5}=\frac{\sqrt{3}-\sqrt{5}}{-2}=\frac{\sqrt{5}-\sqrt{3}}{2}

  • Example: 13−5=\displaystyle \frac{1}{3-\sqrt{5}}=
13−5×3+53+5=3+59−5=3+54\displaystyle \frac{1}{3-\sqrt{5}}\times\frac{3+\sqrt{5}}{3+\sqrt{5}}=\frac{3+\sqrt{5}}{9-5}=\frac{3+\sqrt{5}}{4}

Practice: Radical Expressions

Select the equivalent expressions and state the restrictions on the domain.
a) x−1x−1\displaystyle \frac{x-1}{\sqrt{x}-1}