Laws of indices and surdsEdexcel A-Level Maths: Revision notes

For any rational powers $m$ and $n$ (and non-zero $a$): $a^m\times a^n=a^{m+n},\qquad \frac{a^m}{a^n}=a^{m-n},\qquad (a^m)^n=a^{mn}.$Also $(ab)^n=a^nb^n$, $a^0=1$ and $a^{-n}=\frac{1}{a^n}$. These laws only apply when the bases are the same: $2^3\times3^3$ cannot be combined by adding indices, but it equals $6^3$. Example: $2^{x+3}=2^x\times2^3=8\times2^x$, and $2^{3x-1}=\frac{(2^x)^3}{2}$.

A fractional index means a root: $a^{\frac{m}{n}}=\sqrt[n]{a^m}=\left(\sqrt[n]{a}\right)^m.$A negative index means a reciprocal: $a^{-\frac{m}{n}}=\frac{1}{a^{m/n}}$. Example: $\left(\frac{8}{27}\right)^{-\frac23}=\left(\frac{27}{8}\right)^{\frac23}=\left(\sqrt[3]{\frac{27}{8}}\right)^2=\left(\frac32\right)^2=\frac94$. Take the root first, then the power, because this keeps the numbers small. Also $16^{\frac14}=2$ and $16^{-\frac34}=\frac{1}{8}$.

A surd is a root that cannot be written as a rational number, such as $\sqrt2$ or $\sqrt{18}$. Rules for non-negative numbers: $\sqrt{xy}=\sqrt{x}\times\sqrt{y},\qquad \sqrt{\frac{x}{y}}=\frac{\sqrt x}{\sqrt y},\qquad(\sqrt x)^2=x.$To simplify, take out the largest square factor: $\sqrt{18}=\sqrt{9}\times\sqrt2=3\sqrt2$, and $\sqrt{50}+\sqrt{8}=5\sqrt2+2\sqrt2=7\sqrt2$. Like terms combine, but $\sqrt2+\sqrt3\neq\sqrt5$. Expand brackets as normal: $(2+\sqrt5)^2=4+4\sqrt5+5=9+4\sqrt5$.

To rationalise means to remove surds from the denominator.

• Single surd: multiply top and bottom by it. $\frac{6}{\sqrt3}=\frac{6\sqrt3}{3}=2\sqrt3$.
• Two-term denominator: multiply by the conjugate, which changes the sign between the terms, using $(a+b)(a-b)=a^2-b^2$. Example: $\frac{11+5\sqrt5}{3+\sqrt5}\times\frac{3-\sqrt5}{3-\sqrt5}=\frac{33-11\sqrt5+15\sqrt5-25}{9-5}=\frac{8+4\sqrt5}{4}=2+\sqrt5$. Answers are normally given in the form $a+b\sqrt c$.

The same rules work with letters, for positive $x$ and $y$: $(\sqrt x)^2=x,\qquad\sqrt{xy}=\sqrt x\sqrt y,\qquad(\sqrt x+\sqrt y)(\sqrt x-\sqrt y)=x-y.$So $\frac{1}{\sqrt x-\sqrt y}=\frac{\sqrt x+\sqrt y}{x-y}$. Example: if $\sqrt x-\sqrt y=1$ and $x-y=5$ then $\sqrt x+\sqrt y=\frac{x-y}{\sqrt x-\sqrt y}=5$, and adding gives $\sqrt x=3$, $\sqrt y=2$. Check that answers are in the required form and the denominator is rational.