Math - Laws of Indices

As I stated before, my school is quite slow at teaching Maths (at least thats what I think). Currently in Year 9, we have started algebra - 2 unknown variables. We have learnt the Law of Indices application is very important, not only are its uses limited only to scientific notation, but it also extends to algebra.

Here are the rules for the 6 different laws of indices.

Rule 1: a^m x aⁿ = a^m+n 

The first rule implies when two bases are the same, but indices are different. This cannot occur when the bases are different. The rule for multiplying is to add the two powers together, and not multiply the bases. 

Ex.1) 9² x 9⁵ =9⁷

Ex.2) 7⁶ x 7⁹ = 7¹⁵

What not to do: 4³ x 4⁶ = 16¹⁰ 

Rule 2: a^m ÷ aⁿ= a^m-n

The second rule implies in division, the bases must be the same but indices may be different. When dividing similar bases with different indices, you subtract indices. 

Ex.1) 10² ÷ 10⁷ = 10^{-5 =} 0.00001

Ex.2) 8⁹ ÷ 8³ = 8^{6 =} 262144

What not to do: 9⁹ ÷ 9⁴ = 1⁵ 

Rule 3: (a^m)ⁿ = a^mn 

For this rule, do not times the number in the brackets by the power. You must first times the power inside the brackets by the one on the outside. After multiplying the powers, then you may multiply the base by the power.

Ex.1) (7³)² = (7)^{6 =} 117649

Ex.2) (3⁷)² = (3)¹⁴ = 4782969

Rule 4: a⁰ = 1 (a≠0)

This rule is simple, whatever number the base is does not matter. If the power is 0, the base is out of the equation. The answer to any number to the power of 0 will be 1. 

Ex.1) 9998324230948324⁰ = 1

Ex.2) 5⁰ = 1

Rule 5: a^-m = 1/a^m

This rule implies whenever the power is a negative number. Simply put the base to the power over 1.  

Ex.1) 5^-3 = 1/5³ = 125

Ex.2) 8^-6 = 1/8⁶ = 262144 

Warning: Do not put the negative sign in front of the power.

Rule 6: a^1/m = 

The last rule implies when the power appears in a form of a fraction. 

Ex.1) 9^1/2 = 2√9 = 3

Ex.2) 125^2/3 = (³√125)²  = 5² = 25