A quantity made up of symbols together with operations (+ - \times \div) is called an algebraic expression. We use the laws of indices to simplify expressions involving indices.

The first law: multiplication
If the two terms have the same base (in this case x) and are to be multiplied together their indices are added.

In general: x^m \times x^n = x^{m+n}

The second law: division
If the two terms have the same base (in this case x) and are to be divided their indices are subtracted.

In general: \dfrac{x^m}{x^n}=x^{m-n}

The third law: brackets
If a term with a power is itself raised to a power then the powers are multiplied together.

In general: (x^m)^n = x^{m \times n}

Negative powers
Consider this example: \dfrac{a^2}{a^6} = a^{2-6} = a^{-4}

Also we can show that: \dfrac{a^2}{a^6} = \dfrac{1}{a^4}

So a negative power can be written as a fraction.

In general: x^{-m} = \dfrac{1}{x^m}

Power of zero
The second law of indices helps to explain why anything to the power of zero is equal to one.

We know that anything divided by itself is equal to one. So \dfrac {x^3}{x^3} = 1

Also we know that \dfrac {x^3}{x^3} = x^{3-3} = x^0 = 1

Therefore, we have shown that \dfrac {x^3}{x^3} = x^0 = 1
Here is an explanation of why this is true:

Laws of Indices:

https://www.youtube.com/watch?v=QYjR4eM4mAU

Example 1:

https://www.youtube.com/watch?v=DKVJx1grHQU

Example 2:

https://www.youtube.com/watch?v=YqKRsU1jGto