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

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

Video lesson

Simplify the following expression using the properties of exponents

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