Radical Expressions

Radical Notations: If $ a$ is a real number, $ n$ is a positive integer, and $ a^{1/n}$ is a real number, then

$\displaystyle \sqrt[n]{a}=a^{1/n}$

In the radical $ \sqrt[n]{a}$, the symbol $ \sqrt[n]{}$ is a radical sign, the number $ a$ is the radicand and $ n$ is the index. Usually we denote $ \sqrt[2]{a}$ as $ \sqrt{a}$ If $ a$ is a real number, $ m$ is an integer, $ n$ is a positive integer, and $ a^{m/n}$ is a real number, then

$\displaystyle \sqrt[n]{a^m}=(\sqrt[n]{a})^m=a^{m/n}$

Rules for Radicals: For all real numbers $ a$ and $ b$, and positive integers $ m$ and $ n$ for which the indicated roots are real numbers:

1. Product Rule: $ \sqrt[n]{a}\cdot\sqrt[n]{b} =\sqrt[n]{ab}$
2. Quotient Rule: $ \sqrt[n]{\frac{a}{b}} =\frac{\sqrt[n]{a}}{\sqrt[n]{b}}$, $ b \ne 0$
3. Power Rule: $ \sqrt[m]{\sqrt[n]{a}}=\sqrt[mn]{a}$

Rationalizing Denominators: No denominator contain a radical.

Exs. $ \frac{2}{\sqrt{5}}=\frac{2\sqrt{5}}{5}$, $ \frac{1}{\sqrt{3}-1}=\frac{\sqrt{3}-1}{2}$

Simplified Radicals: An expression with radicals is simplified when all of the following conditions are satisfied.

1. The radicand has no factor raised to a power greater than or equal to the index.

Ex. $ \sqrt[3]{\Big(\frac{7}{11}\Big)^5}= \frac{7}{11} \sqrt[3]{\Big(\frac{7}{11}\Big)^2}=7 \sqrt[3]{\frac{49}{121}}$.

2. The radicand has no fractions.

Exs. $ \sqrt[3]{\frac{49}{121}}=\frac{\sqrt[3]{49}}{\sqrt[3]{121}}$, $ \sqrt[4]{\frac{3}{5}} =\frac{\sqrt[4]{3}}{\sqrt[4]{5}}$.

3. The denominator is rationalized.

Exs. $ \frac{\sqrt[4]{3}}{\sqrt[4]{5}}=\frac{\sqrt[4]{375}}{5}$, $ \frac{\sqrt[3]{49}}{\sqrt[3]{121}}=\frac{\sqrt[3]{539}}{11}$.

4. Exponents in the Radicand and the index of the radical have no common factor.

Ex. $ \sqrt[6]{5^2}=\sqrt[3]{5}$.

5. All indicated operations have been performed (if possible).

Ex. $ \sqrt[3]{\sqrt[3]{6}}=\sqrt[9]{6}$.

Like Radicals: Radicals with the same radicand and the same index.

Ex. $ 2\sqrt[4]{2}$ and $ 3\sqrt[4]{2}$ are like radicals.


Department of Mathematics
Last modified: 2005-08-30