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 $$m \ne n$$.