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:

Joe R. 11 Plus Maths tutor, A Level Maths tutor, 13 Plus  Maths tutor...

1 year ago

Answered by Joe, a GCSE Maths tutor with MyTutor

Still stuck? Get one-to-one help from a personally interviewed subject specialist


£24 /hr

Daniel K.

Degree: Mathematical and Theoretical Physics (Masters) - Oxford, Merton College University

Subjects offered:Maths, Science+ 5 more

Further Mathematics
-Personal Statements-

“Mathematics and Theoretical Physics, University of Oxford. I enjoy sharing my experience and enthusiasm in Maths with those who could do with some help”

£18 /hr

Diana B.

Degree: Chemistry (Masters) - Imperial College London University

Subjects offered:Maths, Chemistry+ 1 more

-Personal Statements-

“Imperial College London student giving Chemistry, Mathematics and Biology lessons”

MyTutor guarantee

£24 /hr

Samuel C.

Degree: Physics (Bachelors) - Durham University

Subjects offered:Maths, Physics+ 2 more

Further Mathematics

“Hi, I'm Sam Crawford. I'm studying Maths and Physics at Durham University, with an offer from Cambridge for next year, and I absolutely love both subjects.”

About the author

Joe R.

Currently unavailable: until 11/08/2016

Degree: Mathematics (Bachelors) - Warwick University

Subjects offered:Maths, .STEP.


“Hello! My name is Joe, I’m a final year Maths student at the University of Warwick, and I am very excited to help you improve your confidence in maths regardless of your starting level. Maths can be one of the most daunting, confusin...”

MyTutor guarantee

You may also like...

Other GCSE Maths questions

expand the brackets (x+5)(x+3) furthermore what are the two values of x

Express 300 as a product of its prime factors.

Solve the equation 3x squared + 4x – 12 = 0 Give your solutions correct to 2 decimal places.

Mark wants to borrow money to buy a car. His bank offers him a loan of £5,000 to be payed back over 3 years at 4% compound interest. a) Work out the interest acquired in the 2nd year. b) In total how much will Mark end up paying back the bank?

View GCSE Maths tutors

We use cookies to improve your site experience. By continuing to use this website, we'll assume that you're OK with this. Dismiss