What are the law of Indices?
A power, or an index, is used to write a product of numbers very compactly. The plural of index is indices.
Example 1:
Simplify 4^5 x 4³. (The ^ symbol = 'the power of')
You can simplify by doing: 4 x 4 x 4 x 4 x 4 x 4 x 4 x 4 = 4^8
Alternatively, you may notice that 5 + 3 = 8, so when multiplying terms with the same base you can add the powers! The video to the right shows you a step by step tutorial. 

Example 2:
Simplify 9^5 ÷ 9³ (The ^ symbol = 'the power of')
To simplify with division you will still need to multiply but instead of multiplying the factors in the next step, you'll need to divide.
e.g. 9 x 9 x 9 x 9 x 9 ÷ 9 x 9 x 9 = 9² You may notice that the powers 5 and 3, when 3 is subtracted by 5, you end up with 2. When dividing terms with the same base, you subtract the powers. The video to the right gives you a step by step tutorial. 

Index Laws 3  raising powers
Example 3:
Simplify (4^3)^4 (The ^ symbol = 'the power of')
To simplify with raising powers you need to multiply the powers e.g. 4 x 4 x 4 x 4 x 4 x 4 x 4 x 4 x 4 x 4 x 4 x 4 = 4^12.
You may notice that there are four sets of three and you can also simplify by doing 4³ x ^4 = 4^12. Video coming soon! 

Index Laws  with algebra (letters)
Example 4:
Simplify 5a³ x 6a^7 (The ^ symbol = 'the power of')
To simplify these type of index, you will need to multiply the number by the letter depending on the power. For example, 5 x a x a x a x 6 x a x a x a x a x a x a x a. The sum will then look like 5 x 6 x a³+^7 which would give you 5 x 6 which is 30 + a = 30a and 3 + 7 = 10 so 30a^10.
You may notice that you can times together the numbers, as a is on both sides it turns into one a and + 7 and 3 together to make 30a^10. 



