C1 - Simplifying a fraction that has a root on the denominator


In this example we are given (√5-1) as our denominator, and we want to simplify this complex fraction into the form a + b√5. 

In order to do this our first step is to elimate the denominator. To do this, we use the rule associated with the Difference of Two Squares. For example, if you were told to factorise and simplify: x 2 - 16, you would get (x-4)(x+4) since when you expand the '-4x' and '+4x' would cancel each other out to leave x2 - 16. 

To return to our question therefore, we need to multiply our fraction by something to eliminate the √5 on the denominator. This something is (√5 +1) as the opposite signs will mean the middle part of our expanded equation +√5-√5 will cancel each other out, and multiplying √5 by itself will eliminate the square root. But we must remember, that whatever we do to the bottom of the fraction we must also do to the top, therefore we must multiply both the numerator and the denominator by (√5 + 1)

So:  (7+√5)/(√5-1) x (√5+1)/(√5+1)

Multiplying the numerators (top of the fractions) together, we get: (7x√5) + (7x1) + (√5x√5) + (√5x1), which equals: 7√5 + 7 + 5 + √5    

By collecting like terms this simplifies to: 8√5 + 12

Multiplying the denominators (bottom of the fractions) together, we get: (√5x√5) + (√5x1) - (1x√5) - (1x1)     which equals: 5 + √5 - √5 - 1      

Again, by collecting like terms this simplifies to: 4

Therefore, we can also write (7+√5)/(√5-1) as:

(8√5 + 12) / 4

But this can be simplified further as both 8, and 12 are divisible by 4, and therefore this can finally be written as: 

2√5 + 3     or     3 + 2√5

SOLUTION:     a = 3      and      b = 2

Claudia M. GCSE Maths tutor, A Level Maths tutor, A Level Economics t...

1 year ago

Answered by Claudia, who has applied to tutor A Level Maths with MyTutor

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


£20 /hr

Larry R.

Degree: Mathematics (Bachelors) - Warwick University

Subjects offered:Maths, Further Mathematics

Further Mathematics

“I am a Mathematics student at Warwick University. My enthusiasm for all things mathematical has no bounds and it's not just Maths that interests me but also the way that people learn it.  I am patient and friendly. Over the last few y...”

£26 /hr

Sonam T.

Degree: Medicine (Bachelors) - Southampton University

Subjects offered:Maths, Chemistry+ 1 more


“Top tutor from the renowned Russell university group, ready to help you improve your grades.”

MyTutor guarantee

£20 /hr

Viktoriya L.

Degree: accounting and finance (Bachelors) - LSE University

Subjects offered:Maths, Russian+ 5 more

-Personal Statements-

“Hello, My name is Viktoriya. I am a first year student at the London school of Economics and Poliical science studying accounting and finance. I am passionate about maths and finance in particular and wish to convey this passion to yo...”

About the author

£20 /hr

Claudia M.

Degree: Economics (Bachelors) - Durham University

Subjects offered:Maths, Economics


“About Me:Hi! I am an economics student at Durham University, whom understands that our economy is volatile and unpredictable, which can make understanding how it works particularly difficult at times. However, so long as we underst...”

MyTutor guarantee

You may also like...

Other A Level Maths questions

What is a parametric equation?

How do I find the coordinates of maximum and minimum turning points of a cubic equation?

When and how do I use integration by parts?

How to gain an inverse function

View A Level 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