Skills for Learning: Maths & Stats - Powers and Logarithms

According to the BEDMAS rule, exponentiation is done after brackets and before multiplication.

So: 3 × 2³ = 3 × 8 = 24 — but (3 × 2)³ = 6³ = 216.

Negative powers denote the reciprocal. The positive powers are calculated first then the reciprocal is taken.

So: 3-2 = 132=19

If $x^a = b$ then $x$ is the $a^{\mathrm{th}}$ root of $b$. For example, if 6² = 36, then 6 is the second root of 36. The second root is usually called the square root. It is written as 6 = $\sqrt{36}$.

The third root is usually called the cube root. For example, 3 = $\sqrt[3]{27}$.

Note also that when −6 is squared we again obtain 36, that is (−6)² = 36. This means that 36 has another square root, −6. Therefore, $\sqrt{36}$ = ±6

Fractional powers can be written as $a^{\frac{m}{n}}$. A fractional power can be broken into two parts: $m^{\mathrm{th}}$ power and $n^{\mathrm{th}}$ root. You can either do the power first then take the root, or alternatively take the root first and then do the power. Therefore:

$a^{\frac{m}{n}}$ = $\sqrt[n]{a^m}$ = ${\sqrt[n]{a}}^{ m}$

So: ${81}^{\frac{3}{4}}$ = ${\left(\sqrt[4]{81}\right)}^3$ = 3³ = 27

The rules of indices can be used to manipulate powers related expressions. The rules of indices are given as below:

$a^n$ x $a^m=a^{n+m}$

$a^n÷a^m=a^{n-m}$

${\left(a^n\right)}^m=a^{nm}$

$a^{-n}=\frac{1}{a^n}$

$a^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$n$}\right.}=\sqrt[n]{a}$

$a^{\raisebox{1ex}{$n$}\!\left/ \!\raisebox{-1ex}{$m$}\right.}={\left(a^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$m$}\right.}\right)}^n={\left(\sqrt[m]{a}\right)}^n$