Surds and Indices

Definition

\[ \sqrt[n]{a} \quad \text{is a surd if } a \text{ is not a perfect nth power.} \]

Multiplication of Surds

\[ \sqrt{a} \times \sqrt{b} = \sqrt{a \times b} \]

Division of Surds

\[ \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \]

Addition and Subtraction of Like Surds

\[ a\sqrt{b} + c\sqrt{b} = (a + c)\sqrt{b} \] \[ a\sqrt{b} - c\sqrt{b} = (a - c)\sqrt{b} \]

Product of Powers Rule

\[ a^m \times a^n = a^{m+n} \]

Quotient of Powers Rule

\[ \frac{a^m}{a^n} = a^{m-n} \]

Power of a Power Rule

\[ (a^m)^n = a^{mn} \]

Power of a Product Rule

\[ (ab)^n = a^n \times b^n \]

Power of a Quotient Rule

\[ \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \]

Zero Exponent Rule

\[ a^0 = 1 \quad \text{(where } a \neq 0\text{)} \]

Negative Exponent Rule

\[ a^{-n} = \frac{1}{a^n} \]

Fractional Exponent Rule

\[ a^{\frac{m}{n}} = \sqrt[n]{a^m} \]