Rationalise the denominator and simplify: 10/3√5 (2 marks)

First of all, you need to read the question really carefully and look at how many marks it is worth. The question is worth two marks and it is asking you to do two things: firstly, you need to ‘rationalise the denominator’, and then you need to ‘simplify’ your answer.
So, what do we mean by rationalising the denominator? Well, the denominator is the number below the line in a fraction. Rationalising means to change a number from being irrational to being rational. An irrational number is a number that cannot be expressed using integers (whole numbers); it goes on and on in a random pattern beyond the decimal point. Let’s look at the fraction in the question: 10/3√5. What we’re aiming for is the denominator to be an integer. To do this, we need to multiply by a number that will √5 rational. So, we would choose to multiply by √5 because √5 X √5 = 5. (√5 is the same as 5^0.5, so we could write this as 5^0.5 X 5^0.5 = 5^1) Remember that whatever we do to the denominator, we have to do to the numerator (top of the fraction) as well. Let’s tackle the numerator and denominator separately. The numerator is 10 and we are multiplying that by √5. 10 X √5 = 10√5. Now, for the denominator. The denominator is 3√5, which we could also write as 3 X √5. So, we are multiplying 3 X √5 X √5. This would give us 3 X 5 which would equal 15. Now to combine these two parts, we have 10√5/15. Now for the second part of the question: simplifying. We could write our fraction as (10 X √5)/15 and then (10/15) X √5. We cannot simplify √5, but we can simplify 10/15. 5 is both a factor of 10 and 15: 5 goes in to 10 twice and 5 goes into 15 three times. Overall, we have (2/3) X √5, but this looks much neater in the form 2√5/3.