Multiplying and dividing radicals in Math isn’t something that is overly hard to get the hang of.
Provided that the radicals involved have the same index n, then we just have to remember one of the properties of radicals.
\sqrt[n]{a} \times \sqrt[n]{b} = \sqrt[n]{a \times b}
a , b , n ∈ \reals
Also, specifically when dealing with square roots.
Generally, \sqrt{a} \times \sqrt{a} \space = \space a,
and a{\sqrt{x}} \times b{\sqrt{y}} \space = \space ab{\sqrt{xy}}.
Multiplying Radicals Examples
(1.1)
2\sqrt{3} \times 4\sqrt{3} \space = \space (2 \times 4) \times {\sqrt{3 \times 3}} \space = \space 8 \times {\sqrt{9}} \space = \space 8 \times 3 \space = \space 24
(1.2)
\sqrt{3} \times \sqrt{5} \space = \space \sqrt{3 \times 5} \space = \space \sqrt{15}
(1.3)
a) \sqrt{20} \times \sqrt{5} \space = \space \sqrt{20 \times 5} \space = \space \sqrt{100} \space = \space 10
b) \sqrt{8} \times \sqrt{12} \space = \space \sqrt{8 \times 12} \space = \space \sqrt{96} \space = \space \sqrt{16 \times 6} \space = \space \sqrt{16} \times \sqrt{6} \space = \space 4\sqrt{6}
c) \sqrt[4]{8} \times \sqrt[4]{12} \space = \space \sqrt[4]{8 \times 12} \space = \space \sqrt[4]{96} \space = \space \sqrt[4]{16 \times 6} \space = \space \sqrt[4]{16} \times \sqrt[4]{6} \space = \space 2\sqrt[4]{6}
(1.4)
(4 \space {\text{--}} \space \sqrt{3})^2 \space = \space (4 \space {\text{--}} \space \sqrt{3})(4 \space {\text{--}} \space \sqrt{3})
= \space 16 \space {\text{--}} \space 4\sqrt{3} \space {\text{--}} \space 4\sqrt{3} + ( \sqrt{3} )^2
= \space 16 {\text{--}} \space 8\sqrt{3} + 3 \space = \space 19 \space {\text{--}} \space 8\sqrt{3}
(1.5)
(\sqrt{x} \space {\text{--}} \space 3)(\sqrt{x} + 4)
= \space (\sqrt{x})^2 + 4\sqrt{x} \space {\text{--}} \space 3\sqrt{x} \space {\text{--}} \space 12
= \space x + \sqrt{x} \space {\text{--}} \space 12
(1.6)
(3\sqrt{x} + 1)(3\sqrt{x} \space {\text{--}} \space 1)
= \space (3\sqrt{x})^2 \space {\text{--}} \space 3\sqrt{x} + 3\sqrt{x} \space {\text{--}} \space 1^2
= \space 9x \space {\text{--}} \space 1
Example (2.5) shows that sometimes when you carry out a radical multiplication, the radical terms are eliminated and there are none present in the result.
Multiplying and Dividing Radicals,
Dividing Examples
Like with multiplying radicals, with dividing radicals we recall a property of radicals.
{\Large{\frac{\sqrt[n]{a}}{\sqrt[n]{b}}}} \space = \space {\Large{\sqrt[n]{\frac{a}{b}}}}
a , b , n ∈ \reals , a \ge 0 \space , \space b \space {\cancel{=}} \space 0 \space , \space n \ge 1
Examples
(2.1)
Simplify \frac{\sqrt[3]32}{\sqrt[3]4}.
Solution
\frac{\sqrt[3]32}{\sqrt[3]4} = \sqrt[3]{\frac{32}{4}} = \sqrt[3]{8} = 2
(2.2)
Simplify \frac{\sqrt[5]486}{\sqrt[5]2}.
Solution
\frac{\sqrt[5]486}{\sqrt[5]2} = \sqrt[5]{\frac{486}{2}} = \sqrt[5]{243} = 3
Dividing radicals examples are often more involved in terms of simplifying than the two examples shown here.
Further examples are introduced on the rationalizing the denominator page.