Laws of Indices (SQA National 5 Maths): Revision Note

$a^1=a$

Anything to the power of 1 is itself

$x^1=x$

$a^0=1$

Anything to the power of 0 is 1

$b^0=1$

$a^m\times a^n=a^{m+n}$

To multiply indices with the same base, add their powers

$\begin{array}{l}{c}^3\times c^2\\ =\left(c\times c\times c\right)\times \left(c\times c\right)\\ =c^5\end{array}$

$a^m÷a^n=\frac{a^m}{a^n}=a^{m-n}$

To divide indices with the same base, subtract their powers

$\begin{array}{l}{d}^5÷d^2\\ =\frac{d\times d\times d\times \cancel{d}\times \cancel{d}}{\cancel{d}\times \cancel{d}}\\ =d^3\end{array}$

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

To raise indices to a new power, multiply their powers

$\begin{array}{l}{\left(e^3\right)}^2\\ =\left(e\times e\times e\right)\times \left(e\times e\times e\right)\\ =e^6\end{array}$

${\left(ab\right)}^m=a^m{b}^m$

To raise a product to a power, apply the power to both numbers, and multiply

$\begin{array}{l}{\left(fg\right)}^2\\ ={\left(f\times g\right)}^2\\ =f^2\times g^2\\ =f^2{g}^2\end{array}$

${\left(\frac{a}{b}\right)}^m=\frac{a^m}{b^m}$

To raise a fraction to a power, apply the power to both the numerator and denominator

${\left(\frac{h}{i}\right)}^2=\frac{h^2}{i^2}$

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

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

A negative power is the reciprocal

$j^{-1}=\frac{1}{j}$

$k^{-3}=\frac{1}{k^3}$

${\left(\frac{a}{b}\right)}^{-n}={\left(\frac{b}{a}\right)}^n=\frac{b^n}{a^n}$

A fraction to a negative power, is the reciprocal of the fraction, to the positive power

${\left(\frac{l}{m}\right)}^{-3}={\left(\frac{m}{l}\right)}^3=\frac{m^3}{l^3}$

$a^{\frac{1}{n}}=\sqrt[n]{a}$

The fractional power $\frac{1}{n}$ is the n^th root ( )

$n^{\frac{1}{2}}=\sqrt{n}$

$p^{\frac{1}{3}}=\sqrt[3]{p}$

$\begin{array}{l}{a}^{-\frac{1}{n}}={\left(a^{\frac{1}{n}}\right)}^{-1}\\ ={\left(\sqrt[n]{a}\right)}^{-1}=\frac{1}{\sqrt[n]{a}}\end{array}$

A negative, fractional power is one over a root

$q^{-\frac{1}{2}}=\frac{1}{\sqrt{q}}$

$r^{-\frac{1}{3}}=\frac{1}{\sqrt[3]{r}}$

$\begin{array}{rcl}{a}^{\frac{m}{n}}& =& a^{\frac{1}{n}\times m}\\ & =& {\left(a^{\frac{1}{n}}\right)}^m={\left(\sqrt[n]{a}\right)}^m\\ \mathrm{or}& =& {\left(a^m\right)}^{\frac{1}{n}}=\sqrt[n]{a^m}\end{array}$

The fractional power $\frac{m}{n}$ is the n^th root all to the power m, , or the n^th root of the power m, (both are the same)

$s^{\frac{2}{3}}={\left(s^{\frac{1}{3}}\right)}^2={\left(\sqrt[3]{s}\right)}^2$

$s^{\frac{2}{3}}={\left(s^2\right)}^{\frac{1}{3}}=\sqrt[3]{s^2}$