Indices and surds

Laws of indices

• $x^m\times x^n=x^{m+n}$ · $\dfrac{x^m}{x^n}=x^{m-n}$ · $(x^m)^n=x^{mn}$
• $x^{-n}=\dfrac{1}{x^n}$ — a negative index gives a reciprocal, not a negative answer
• $x^{m/n}=\sqrt[n]{x^m}$ — the denominator is the root, the numerator is the power
• $x^0=1$ for any $x\ne0$

$\left(\dfrac{8}{27}\right)^{2/3}=\left(\sqrt[3]{\dfrac{8}{27}}\right)^2=\left(\dfrac23\right)^2=\dfrac49$

Surds

Simplify by pulling out square factors: $\sqrt{18}=\sqrt{9\times2}=3\sqrt2$. Only like surds can be added.

Rationalising

Single surd: multiply top and bottom by that surd. Binomial: multiply by the conjugate — $\dfrac{1}{\sqrt7-2}\times\dfrac{\sqrt7+2}{\sqrt7+2}=\dfrac{\sqrt7+2}{3}$.