Rationalise 4/(√5 + 1) Step by Step

Step by step

The question

Question: rationalise 4 over (√5 + 1).

4√5 + 1

Rationalise means: write it with no root at the bottom.

1 Find the partner

With one root at the bottom, we multiplied by that root.

With two terms, we use the partner: the same two terms, with the other sign.

The partner of √5 + 1 is √5 − 1.

partner: √5 + 1 → √5 − 1

So we multiply by √5 − 1 over √5 − 1.

√5 − 1 over √5 − 1 is 1, and times 1 changes nothing.

= 4√5 + 1 × √5 − 1√5 − 1

Careful!

Times √5 alone does not work here.

√5 + 1 times √5 still has a root in it.

Careful: not × √5√5 ✗

Top times top, bottom times bottom.

= 4 × (√5 − 1)(√5 + 1)(√5 − 1)

2 Bottom: difference of squares

Why the partner?

a plus b, times a minus b, is a squared minus b squared.

A root squared has no root.

(a + b)(a − b) = a² − b²

On the bottom: (√5) squared minus 1 squared.

√5 squared is 5.

So it is 5 minus 1: 4.

No root at the bottom now.

(√5)² − 1² = 5 − 1 = 4

So the bottom is 4.

= 4 × (√5 − 1)4

3 Top: open the brackets

On top: 4 times every part of the bracket.

That is 4√5 − 4.

4 × (√5 − 1) = 4√5 − 4

So we get (4√5 − 4)/4.

= 4√5 − 44

4 Simplify

4, 4 and 4 can all be divided by 4.

4, 4, 4: all ÷ 4

Over 1 is just the top: √5 − 1.

= √5 − 1

5 Check

Let's check with a calculator.

The start is about 1.24.

Start: 4√5 + 1 ≈ 1.24

The answer is about 1.24 too.

So it is right!

Answer: √5 − 1 ≈ 1.24

Let's look back.

First: find the partner.

Then: bottom: difference of squares.

Then: top: open the brackets.

Then: simplify.

Last: check.

Answer