Rationalise 1/(√3 + 1) Step by Step

Step by step

The question

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

1√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.

= 1√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.

= 1 × (√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.

= 1 × (√3 − 1)2

3 Top: open the brackets

On top: 1 times every part of the bracket.

That is √3 − 1.

1 × (√3 − 1) = √3 − 1

So we get (√3 − 1)/2.

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

= √3 − 12

4 Check

Let's check with a calculator.

The start is about 0.37.

Start: 1√3 + 1 ≈ 0.37

The answer is about 0.37 too.

So it is right!

Answer: √3 − 12 ≈ 0.37

Let's look back.

First: find the partner.

Then: bottom: difference of squares.

Then: top: open the brackets.

Last: check.

Answer