Exponents & Surds

Everything for this chapter, in one place. $a, b > 0$ unless stated otherwise.

The laws

Solving an exponential equation


Same base both sides	equate the exponents — valid when the base is not $0$ , $1$ or $-1$
Different bases	prime-factorise both sides, then match each prime separately
$a^{2x}$ and $a^x$ both present	substitute $t = a^x$ and solve the quadratic
Three terms, middle one doubled	look for $A^2 - 2AB + B^2 = (A-B)^2$

$f(x)^{g(x)} = 1$ has three doors

Surds


Products	$\sqrt{a}\sqrt{b} = \sqrt{ab}$
Rationalising	$\dfrac{1}{\sqrt{a}+\sqrt{b}} = \dfrac{\sqrt{a}-\sqrt{b}}{a-b}$
Squaring a surd pair	$(a+b\sqrt{n})^2 = a^2 + nb^2 + 2ab\sqrt{n}$
Denesting	match the rational parts and the $\sqrt{\ }$ parts separately
If $A+B$ is known and $A^2-B^2$ is known	$A - B = \dfrac{A^2-B^2}{A+B}$ , then add to get $A$

Climbing the $x^n + \frac{1}{x^n}$ ladder


Up one rung	$\left(x^n + \dfrac{1}{x^n}\right)^2 = x^{2n} + \dfrac{1}{x^{2n}} + 2$
Cubes	$x^3 + \dfrac{1}{x^3} = \left(x+\dfrac{1}{x}\right)^3 - 3\left(x+\dfrac{1}{x}\right)$
Combining two rungs	$\left(x^a + \dfrac{1}{x^a}\right)\left(x^b + \dfrac{1}{x^b}\right) = x^{a+b} + \dfrac{1}{x^{a+b}} + x^{a-b} + \dfrac{1}{x^{a-b}}$

Sizes worth knowing — for comparison questions

Formula sheet