AS Maths Core One Indices

When multiplying numbers with powers you can add the powers together.

eg. X^a x X^b = X^a+b

But only when the base numbers are the same.

eg. X^a x Y^b X Not Possible to add the powers here

Simplify the following....

2^7 x 2^7 = 2^14

3^6 x 3^7 = 3^13

5^6 x 7^6= Cannot add the powers here because the base numbers are not the same.

a^8 x a^6 = a^14

When dividing numbers with powers you can subtract the powers.

eg. X^a ÷ X^b = X^a-b

But only when the base numbers are the same.

eg. X^a x Y^b X Not Possible

Simplify the following...

3⁶ ÷ 3^2 = 3^4

5^5 ÷5^6= 5-1

t^8 ÷ t^4= t^4

n^3 ÷ m^5= Cannot subtract the powers here because the base numbers are different

When numbers with powers are in a bracket and there's a power outside the bracket too, you canmultiply the powers.

eg. (X^{a)^b= Xaxb}

When there's a term in the bracket in front of the Xyou must multiply that by the indice outside the bracket too.

eg. (2x^a)^b= 2^b x X^axb

Simplify the following...

(a^{4)^5 = a^20}

(t-3)^{4= t-12}

(3X^5)^4 = 81X^20

4^3 = 2 ^6

This is because 4^3 can also be written as (2^2)^3 because 2^2 is the same as 2 x 2 =4.

Therefore (2^2)^3 = 2^6 because you can multiply the powers.

eg. 8^2 = (2^3)^2 = 2^6

More complex example...

16² x 8³ ÷ 32²

Firstly you can write 16^2 as (2^4)^2

Then 8^3 as (2^3)^3

And 32^2 as (2^5)^2

So far we have (2^4)^2 x (2^3)^3 ÷ (2^5)2

Using the power on power rule we can simplify this to:

2^8 x 2^9 ÷ 2^10

We can then use the adding power rule to simplify 2^8 x 2^9 =2^17

So 2^17 ÷ 2^10

Then we use the subtracting powers rule to get 2^7

Negative powers indicate that the number can be written as a fraction for example

x^{-1 could be written as 1/x}

or

x-2 could be written as 1/x²

or

x-3 could be written as 1/x^3

Write the following as fractions...

2^{3 = 1/23 = 1/8}

5-1 = 1/5

10-4 = 1/10^4 = 1/10000

When a number is to a fractional power there is an alternative way you can write it for example:

x^1/2 = √x The square root of x to the power of one

or

x^1/3 = 3√x The cube root of x to the power of one

The denominator determines what root it is and the numerator determines what power the base number is to.

Solve the following

9^1/2 = √9 = 3

27^1/3 = 3√27 = 3