Mathematics » Exponents and Surds » Rational Exponents And Surds

# Simplification of Surds

## 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:

1. $$\sqrt[n]{a} \times \sqrt[n]{b} = \sqrt[n]{ab}$$
2. $$\sqrt[n]{\dfrac{a}{b}} = \dfrac{\sqrt[n]{a}}{\sqrt[n]{b}}$$
1. \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*}
2. \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:

1. $$\sqrt{2} \times \sqrt{32} = \sqrt{2 \times 32} = \sqrt{64} = 8$$

2. $$\dfrac{\sqrt[3]{24}}{\sqrt[3]{3}} = \sqrt[3]{\dfrac{24}{3}} = \sqrt[3]{8} = 2$$

3. $$\sqrt{\sqrt{81}} = \sqrt[4]{81} = \sqrt[4]{3^4} = 3$$

## 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 $$m \ne n$$.