Lesson 13 · from §95 of the 1910 text
Division of Radicals
Division of radicals
Direct division. Divide coefficient by coefficient and radicand by radicand, then simplify:
Example 1. \sqrt6 \div \sqrt2 = \dfrac{\sqrt6}{\sqrt2} = \sqrt{\dfrac62} = \sqrt3.
Example 2. 6\sqrt5 \div 3\sqrt3 = 2\sqrt{\dfrac53}, which becomes \dfrac23\sqrt{15}.
Example 3. Radicals of different orders are first reduced to the same order: 8\sqrt{12} \div 2\sqrt[3]{6} = \dfrac{8(12)^{1/2}{2(6)^{1/3} = \dfrac{4(12)^{3/6}{(6)^{2/6} = 4\sqrt[6]{\dfrac{12 \cdot 12 \cdot 12}{6 \cdot 6}} = 4\sqrt[6]{48}.
Division by a rationalizing factor. If the divisor is a monomial surd it is always possible, and often far easier, to multiply both terms of the fraction by a rationalizing factor of the divisor:
Example 4. 4 \div \sqrt3 = \dfrac{4}{\sqrt3} = \dfrac{4\sqrt3}{\sqrt3\,\sqrt3} = \dfrac{4\sqrt3}{3}.
Example 5. 6 \div \sqrt[3]3 = \dfrac{6\sqrt[3]9}{\sqrt[3]3 \cdot \sqrt[3]9} = \dfrac{6\sqrt[3]9}{3} = 2\sqrt[3]9.
Example 6. \sqrt3 \div \sqrt[3]2 = \dfrac{\sqrt3 \cdot \sqrt[3]4}{\sqrt[3]2 \cdot \sqrt[3]4} = \dfrac{3^{1/2} \cdot 4^{1/3}{2} = \dfrac{3^{3/6} \cdot 4^{2/6}{2} = \tfrac12\sqrt[6]{3^3 \cdot 4^2} = \tfrac12\sqrt[6]{432}.
Why bother? To find the value of 2 \div \sqrt3 directly we must extract \sqrt3 and then divide by a long decimal; as \dfrac{2\sqrt3}{3} there is only one long operation.
How to type radicals
- Square root:
sqrt(12)or√12; a coefficient goes in front:2sqrt(3)or2√3. - Cube root:
cbrt(5)or∛5. Any other root:root(x, 4)for \sqrt[4]{x},root(2, 6)for \sqrt[6]{2}. - Put the whole radicand in parentheses when it is more than one number or letter:
sqrt(2x),cbrt(a + b),sqrt(2 - sqrt(2)). - Fractional exponents use
^with parentheses:x^(2/3). - Letters under a radical sign stand for positive numbers, so \sqrt{x^2} = x.
Leave no radical in a denominator: \dfrac{4}{\sqrt3} should be written \dfrac{4\sqrt3}{3}, typed 4sqrt(3)/3.
Exercises
How to type math
Use ^ for powers (a^2), / for fractions ((a+b)/2), sqrt(x) for square roots, and cbrt(x) for cube roots. You can leave out the multiplication sign: 3ab means 3 \cdot a \cdot b. Mixed numbers like 4 1/2 work too. The preview under each box shows how your answer was read.
Perform the indicated division:
\sqrt{10} \div \sqrt2
(18)^{1/2} \div (3)^{1/2}
\dfrac{\sqrt6}{\sqrt{18}}
6 \div 2\sqrt2
\dfrac{3\sqrt{12}}{\sqrt6}
8\sqrt{15} \div 4\sqrt5
\dfrac{10}{2\sqrt5}
\dfrac{4\sqrt{10}}{8\sqrt5}
8 \div 4\sqrt3
\dfrac{3\sqrt2}{15\sqrt8}
\left(\sqrt6 + \sqrt{18}\right) \div 3\sqrt2
\left(\sqrt{12} - \sqrt{24}\right) \div 2\sqrt3
\dfrac{6\sqrt{10} + 4\sqrt{15} - \sqrt{20}}{2\sqrt5}
\left(12 + \sqrt3 + \sqrt5\right) \div \sqrt6
\dfrac{\sqrt6 - \sqrt9 + 18}{2\sqrt2}
\left(\sqrt5 + 2\right) \div \sqrt{125}
\sqrt{\tfrac{32}{10}} \div \sqrt{\tfrac45}
(xy)^{1/2} \div x^{1/2}
a\sqrt{bc} \div d\sqrt c
\dfrac{a^2\sqrt c}{a\sqrt{bc}}
\sqrt[3]{16} \div \sqrt[3]4
8\sqrt[3]{125} \div 4\sqrt[3]{25}
\dfrac{2\sqrt[3]5}{3\sqrt[3]4}
a9^{1/3} \div b8^{1/3}
\sqrt[3]{\tfrac18} \div \sqrt[3]{\tfrac14}
\sqrt[3]4 \div \sqrt[2]2
\sqrt2 \div \sqrt[3]4
4^{1/3} \div 6^{1/2}
6^{1/2} \div 4^{1/3}
\sqrt[5]{27} \div \sqrt{27}
\sqrt[4]{\tfrac18} \div \sqrt[3]{\tfrac12}
\sqrt[6]{\dfrac{a^2}{72a}} \div \sqrt[3]{\dfrac a{81}}