## Simplification of Surds

We have seen in previous examples and exercises that rational exponents are closely related to surds. It is often useful to write a surd in exponential notation as it allows us to use the exponential laws.

The additional laws listed below make simplifying surds easier:

- \(\sqrt[n]{a}\sqrt[n]{b} = \sqrt[n]{ab}\)
- \(\sqrt[n]{\dfrac{a}{b}} = \dfrac{\sqrt[n]{a}}{\sqrt[n]{b}}\)
- \(\sqrt[m]{\sqrt[n]{a}} = \sqrt[mn]{a}\)
- \(\sqrt[n]{a^m} = a^{\frac{m}{n}}\)
- \(( \sqrt[n]{a} )^m = a^{\frac{m}{n}}\)

## Example

### Question

Show that:

- \(\sqrt[n]{a} \times \sqrt[n]{b} = \sqrt[n]{ab}\)
- \(\sqrt[n]{\dfrac{a}{b}} = \dfrac{\sqrt[n]{a}}{\sqrt[n]{b}}\)

- \begin{align*} \sqrt[n]{a} \times \sqrt[n]{b} &= a^{\frac{1}{n}} \times b^{\frac{1}{n}} \\ &= (ab)^{\frac{1}{n}} \\ &= \sqrt[n]{ab} \end{align*}
- \begin{align*} \sqrt[n]{\cfrac{a}{b}} &= ( \cfrac{a}{b} )^{\frac{1}{n}} \\ &= \dfrac{a^{\frac{1}{n}}}{b^{\frac{1}{n}}} \\ &= \dfrac{\sqrt[n]{a}}{\sqrt[n]{b}} \end{align*}

Examples:

\(\sqrt{2} \times \sqrt{32} = \sqrt{2 \times 32} = \sqrt{64} = 8\)

\(\dfrac{\sqrt[3]{24}}{\sqrt[3]{3}} = \sqrt[3]{\dfrac{24}{3}} = \sqrt[3]{8} = 2\)

\(\sqrt{\sqrt{81}} = \sqrt[4]{81} = \sqrt[4]{3^4} = 3\)

## Like and Unlike Surds

Two surds \(\sqrt[m]{a}\) and \(\sqrt[n]{b}\) are like surds if \(m=n\), otherwise they are called unlike surds. For example, \(\sqrt{\cfrac{1}{3}}\) and \(-\sqrt{61}\) are like surds because \(m = n = 2\). Examples of unlike surds are \(\sqrt[3]{5} \text{ and } \sqrt[5]{7y^3}\) since \(m \ne n\).