Solving equations with indexes

If you see something like 0.2^t in the equation and you have to solve for t, then the trick is to make the right hand side look like 0.2 to the power of something as well. For this it helps to recognise the lower powers of the smaller integers.

(NOTE: these comments are aimed at GCSE foundation: if you know about logarithms then there is a much more direct and general way to solve these equations. Just take logarithms of both sides of the equation).

Example:

to solve 5 x 0.2^t = 6.4 x 10^-5

First divide by 5 to get rid of it on the left

0.2^t = 1.28 x 10^-5

recognise that 128 = 2⁷ to get

0.2^t = 2⁷/100 x 10^-5

Dividing by 100 is the same as multiplying by 10^-2

0.2^t = 2⁷ x 10^-2 x 10^-5

Multiplying two powers is as easy as adding the indices.

0.2^t = 2⁷ x 10^-7

A negative power is 1 over the positive power, so this is

0.2^t = 2⁷ x (1/10)⁷

Now, if two things multiplied together are both to the power of 7 then you can simplify:

0.2^t = (2/10)⁷ = 0.2⁷

Now you have the same thing raised to a power on each side, so equating the indices gives t=7.