# Properties of exponents

In earlier chapters we introduced powers.

$x^{3}=x\cdot x\cdot x$

There are a couple of operations you can do on powers and we will introduce them now.

We can multiply powers with the same base

$x^{4}\cdot x^{2}=\left (x\cdot x\cdot x\cdot x \right )\cdot \left ( x\cdot x \right )=x^{6}$

This is an example of the product of powers property tells us that when you multiply powers with the same base you just have to add the exponents.

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

We can raise a power to a power

$\left ( x^{2} \right )^{4}= \left (x\cdot x \right )\cdot \left (x\cdot x \right ) \cdot \left ( x\cdot x \right )\cdot \left ( x\cdot x \right )=x^{8}$

This is called the power of a power property and says that to find a power of a power you just have to multiply the exponents.

When you raise a product to a power you raise each factor with a power

$\left (xy \right )^{2}= \left ( xy \right )\cdot \left ( xy \right )= \left ( x\cdot x \right )\cdot \left ( y\cdot y \right )=x^{2}y^{2}$

This is called the power of a product property

$\left (xy \right )^{a}= x^{a}y^{a}$

As well as we could multiply powers we can divide powers.

$\frac{x^{4}}{x^{2}}=\frac{x\cdot x\cdot {\color{red} \not}{x}\cdot {\color{red} \not}{x}}{{\color{red} \not}{x}\cdot {\color{red} \not}{x}}=x^{2}$

This is an example of the quotient of powers property and tells us that when you divide powers with the same base you just have to subtract the exponents.

$\frac{x^{a}}{x^{b}}=x^{a-b},\: \: x\neq 0$

When you raise a quotient to a power you raise both the numerator and the denominator to the power.

$\left (\frac{x}{y} \right )^{2}=\frac{x}{y}\cdot \frac{x}{y}=\frac{x\cdot x}{y\cdot y}=\frac{x^{2}}{y^{2}}$

This is called the power of a quotient power

$\left (\frac{x}{y} \right )^{a}=\frac{x^{a}}{y^{a}},\: \: y\neq 0$

When you raise a number to a zero power you'll always get 1.

$1=\frac{x^{a}}{x^{a}}=x^{a-a}=x^{0}$

$x^{0}=1,\: \: x\neq 0$

Negative exponents are the reciprocals of the positive exponents.

$x^{-a}=\frac{1}{x^{a}},\: \: x\neq 0$

$x^{a}=\frac{1}{x^{-a}},\: \: x\neq 0$

The same properties of exponents apply for both positive and negative exponents.

In earlier chapters we talked about the square root as well. The square root of a number x is the same as x raised to the 0.5th power

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

## Video lesson

Simplify the following expression using the properties of exponents

$\frac{( 7^{5}) ^{10}\cdot 7^{200}}{\left ( 7^{-2} \right )^{30}}$