Rationalise 3/(2√2 - 1) Step by Step

Step by step

The question

Question: rationalise 3/(2√2 − 1).

32√2 − 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 2√2 − 1 is 2√2 + 1.

partner: 2√2 − 1 → 2√2 + 1

So we multiply by 2√2 + 1/2√2 + 1.

2√2 + 1/2√2 + 1 is 1, and times 1 changes nothing.

= 32√2 − 1 × 2√2 + 12√2 + 1

Careful!

Times 2√2 alone does not work here.

2√2 − 1 times √2 still has a root in it.

Careful: not × 2√22√2 ✗

Top times top, bottom times bottom.

= 3 × (2√2 + 1)(2√2 − 1)(2√2 + 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: (2√2) squared minus 1 squared.

2√2 squared is 8: 2 squared times 2.

So it is 8 minus 1: 7.

No root at the bottom now.

(2√2)² − 1² = 8 − 1 = 7

So the bottom is 7.

= 3 × (2√2 + 1)7

3 Top: open the brackets

On top: 3 times every part of the bracket.

That is 3 + 6√2.

3 × (2√2 + 1) = 3 + 6√2

So we get (3 + 6√2)/7.

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

= 3 + 6√27

4 Check

Let's check with a calculator.

The start is about 1.64.

Start: 32√2 − 1 ≈ 1.64

The answer is about 1.64 too.

So it is right!

Answer: 3 + 6√27 ≈ 1.64

Let's look back.

First: find the partner.

Then: bottom: difference of squares.

Then: top: open the brackets.

Last: check.

Answer