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Work out the value of 125 to the power of -2/3.

We want to find 125-2/3. To do this, we will need to use lots of power rules that we have learnt.

Firstly, when we put a power to another power, we multiply the two powers. For example, (23)4  (2 to the power of 3, ALL to the power of 4), is equal to 23x4 = 212.

We are going to use this rule in reverse. We can rewrite our power, -2/3, as the product of numbers that we know how to work with as powers.

Examples of powers that we know how to use, are:

Positive integers, such as 2 or 3: we just multiply the number by itself this many times, as usual. 
-1 : we put 'one over' the number. For example, 3-1 = 1/3.
Fractional powers, of the form 1/n : we take the n-th root. For example, 41/2 = square root of 4 = 2.

Firstly, we can factor out the negative to make things easier for ourselves, as we know how to work with -1 as a power. So, -2/3 = -1 x 2/3.

Secondly, we know how to use fractional powers, if they have a 1 in the numerator. So, 2/3 = 2 x 1/3.

Putting this all together, we have -2/3 = -1 * 2 * 1/3.

Back to our original question now!

We want to find 125-2/3 which we can now rewrite using our 'power of a power' rule in reverse, to get ((1251/3)2)-1. Because we can multiply numbers in any order and still get the same result, we can apply these powers in any order. But, some orders are often easier than others. I like to do negative powers last, because then we don't need to work with fractions. Also, here, I have noticed an interesting property about the number 125 - it's a cube number! Specifically, 5 cubed! So I will do power of 1/3, which is a cube root, first, to make things nice and simple for myself.

Now we just need to do our powers, one by one, to complete the question.

1251/3 = cube root of 125 = 5.
52 = 25.
25-1 = 1/25.

And we have our answer, 1/25!

If any of this has been confusing, I'd be happy to explain further!
If want to revise the rules in your own time, the BBC website has a great resource on them  here: http://www.bbc.co.uk/education/topics/zxkw2hv.

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