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Surds and Indices Formulas
Surds
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
\[
a\sqrt{b} + c\sqrt{b} = (a + c)\sqrt{b}
\]
\[
a\sqrt{b} - c\sqrt{b} = (a - c)\sqrt{b}
\]
Indices
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
\[
a^{\frac{m}{n}} = \sqrt[n]{a^m}
\]
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