Rationalise 3/(√3 - 1) Step by Step

Step by step

The question

Question: rationalise 3 over (√3 − 1).

3√3 − 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 √3 − 1 is √3 + 1.

partner: √3 − 1 → √3 + 1

So we multiply by √3 + 1 over √3 + 1.

√3 + 1 over √3 + 1 is 1, and times 1 changes nothing.

= 3√3 − 1 × √3 + 1√3 + 1

Careful!

Times √3 alone does not work here.

√3 − 1 times √3 still has a root in it.

Careful: not × √3√3 ✗

Top times top, bottom times bottom.

= 3 × (√3 + 1)(√3 − 1)(√3 + 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: (√3) squared minus 1 squared.

√3 squared is 3.

So it is 3 minus 1: 2.

No root at the bottom now.

(√3)² − 1² = 3 − 1 = 2

So the bottom is 2.

= 3 × (√3 + 1)2

3 Top: open the brackets

On top: 3 times every part of the bracket.

That is 3 + 3√3.

3 × (√3 + 1) = 3 + 3√3

So we get (3 + 3√3)/2.

The numbers can't be cancelled, so we are done.

= 3 + 3√32

4 Check

Let's check with a calculator.

The start is about 4.10.

Start: 3√3 − 1 ≈ 4.10

The answer is about 4.10 too.

So it is right!

Answer: 3 + 3√32 ≈ 4.10

Let's look back.

First: find the partner.

Then: bottom: difference of squares.

Then: top: open the brackets.

Last: check.

Answer