# Exponents

A quick reference to exponents:

Two basic rules:

$x^0=1$

Any number to the power of zero is just one.

Otherwise the (integer) exponent indicates how many times to multiply the number times itself:

$x^n={x_1}\times{x_2}\times\ldots\times{x_n}$

The first rule covers the absurd case of trying to multiply a number times itself zero times. The second rule means that any number to the power of one is that number:

$x^1=x$

Note that:

$x^2=x^{(1+1)}={x^1}\times{x^1}={x}\times{x}$

Which also demonstrates that a number to the power of one is just that number. It also shows how the second rule could be derived from the rule about an exponent of one.

The second rule, put formally, is:

$\displaystyle\prod^{n}_{i=1}x_i$

A third, very important, rule:

$x^{a+b}={x^a}\times{x^b}$

Adding exponents is the same as multiplying the base number. Note that subtracting just means a negative exponent:

$x^{a-b}=x^{a+(-b)}={x^a}\times{x^{-b}}$

The third rule leads to two other important identities. Firstly:

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

Because:

${x^n}\times{x^{-n}}={x^{n-n}}={x^0}=1$

Anything to the power of zero is equal to one. Therefore:

${x^n}\times{x^{-n}}=1$

And dividing both sides by xn:

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

Secondly, the third rule gives us:

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

Because:

${x^1}=x^{\frac{1}{2}+\frac{1}{2}}={x^\frac{1}{2}}\times{x^\frac{1}{2}}=x$

Therefore:

$({x^\frac{1}{2}})^2=x$

Taking the square root of both sides gives us:

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

Similar constructions can prove cubes and higher powers.

Finally:

$x^{{a}\times{b}}=(x^a)^b$

Multiplying exponents is the same as raising the first result to the second exponent.