Index laws

Laws that define how to deal with exponents in algebra.

Multiplication

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

Division

$a^{m} \div a^{n} = a^{m-n}$

Powers

$(a^{4})^{3} = a^{4} \times a^{4} \times a^{4}$
$= a^{4+4+4}$ $= a^{4 \times 3} = a^{12}$

$(a^{m})^{n} = a^{mn}$
$(ab)^{n} = a^{n} b^{n}$

$x^{m+1} \times x^{m-1} = x^{2m}$

Fractions

Since $a^{\frac 1 2} \times a^{\frac 1 2} = a^{1} = a$ it follows that $a^{\frac 1 2} = \sqrt[2]{a}$.

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

$a^{0.25} = a^{\frac 1 4} = \sqrt[4]{a}$

These may sometimes require to be simplified:

$x^{\frac{31}{12}} = \sqrt[12]{x^{31}}$

Since $12$ goes into $31$ two times, and the remainder is $7$:

$\Rarr \sqrt[12]{x^{24}} \times \sqrt[12]{x^{7}}$

And since the former factor simplifies further into $x^{2}$, it follows that

$\sqrt[12]{x^{31}} = x^{2}\sqrt[12]{x^{7}}$

Zero Indices

$a^{n} \div a^{n} = a^{n-n} = a^{0}$

So $a^{0} = a^{n} \div a^{n}$, also, since every number divided by itself results in 1, it follows that $a^{0} = 1$.

$0^{0} =$ undefined

Negative Indices

$a^{-1}$ is equal to the reciprocal of $a$

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

$2a^{-3} = \frac{2}{a^{3}}$

$a^{-\frac 1 2 } = \frac{1}{\sqrt[2]{a}}$

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

$(\frac 1 x)^{-1} = \frac x 1 = x$

$(\frac a b)^{-c} = \frac{b^{c}}{a^{c}}$

Notes

Careful with negative numbers:

$-4^2 \ne (-4)^2$ because
$-4^2 = -(4^2)$