Operations with radicals

For radicals that have the same index, rules analogous to those for powers hold.

Property — Operations with radicals

• Product: $\sqrt[n]{a}\cdot\sqrt[n]{b} = \sqrt[n]{ab}$  (same index!)
• Quotient: $\dfrac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\dfrac{a}{b}}$
• Power: $\bigl(\sqrt[n]{a}\bigr)^m = \sqrt[n]{a^m} = a^{\frac{m}{n}}$
• Inverse: $\bigl(\sqrt[n]{a}\bigr)^n = a$

Caution

You CANNOT add radicals with different arguments! $\sqrt{2}+\sqrt{3} \neq \sqrt{5}$ Indeed $(\sqrt{3}+\sqrt{2})^2 = 3+2\sqrt{6}+2 = 5+2\sqrt{6} \neq 5$.

Links

Topics: Radicals
Concepts: Radical
Skills: Simplifying · Using formulae