Indices & Surds

At a glance 12 formulas

1. 1Same base$a^m \times a^n = a^{m+n}$$\dfrac{a^m}{a^n} = a^{m-n}$
2. 2Power of a power$(a^m)^n = a^{mn}$$(ab)^n = a^n b^n$
3. 3Zero and negative powers$a^0 = 1,\ a^{-n} = \dfrac{1}{a^n}$$\left(\dfrac{a}{b}\right)^{-n} = \left(\dfrac{b}{a}\right)^{n}$
4. 4Fractional powers$a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^{m}$$\sqrt{a} \times \sqrt{b} = \sqrt{ab}$
5. 5Equal powers$a^x = a^y \Rightarrow x = y$$\text{when } a > 0,\ a \ne 1$
6. 6Rationalise$\dfrac{1}{\sqrt{a} + \sqrt{b}} = \dfrac{\sqrt{a} - \sqrt{b}}{a - b}$$\dfrac{1}{\sqrt{n} + \sqrt{n+1}} = \sqrt{n+1} - \sqrt{n}$
7. 7Root of a surd$\sqrt{a + b + 2\sqrt{ab}} = \sqrt{a} + \sqrt{b}$$\sqrt{a + b - 2\sqrt{ab}} = \sqrt{a} - \sqrt{b},\ a > b$
8. 8Reciprocal surds$a^2 - b = 1 \Rightarrow \dfrac{1}{a + \sqrt{b}} = a - \sqrt{b}$$\text{e.g. } (2 + \sqrt{3})(2 - \sqrt{3}) = 1$
9. 9Factor the smallest powerTake out the smallest power: $3^{n+2} + 3^{n} = 3^{n}(9 + 1)$. The unknown power then stands alone, or cancels.
10. 10Compare rootsRaise every root to the LCM of the root indices, then compare whole numbers: $\sqrt{2}$ vs $\sqrt[3]{3}$ becomes $2^3 = 8$ vs $3^2 = 9$.
11. 11Common value$a^x = b^y = k \Rightarrow a = k^{\frac{1}{x}},\ b = k^{\frac{1}{y}}$$\text{if also } c^z = k \text{ and } c = a^p b^q:\ \tfrac{1}{z} = \tfrac{p}{x} + \tfrac{q}{y}$
12. 12Power equal to 1$a^b = 1$ in three cases: $a = 1$; $b = 0$ with $a \ne 0$; $a = -1$ with $b$ an even integer. Check all three and count repeats once.