Rationalise 2/(1 - √2) Step by Step

Step by step

The question

Question: rationalise 2/(1 − √2).

21 − √2

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 1 − √2 is 1 + √2.

partner: 1 − √2 → 1 + √2

So we multiply by 1 + √2/1 + √2.

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

= 21 − √2 × 1 + √21 + √2

Careful!

Times √2 alone does not work here.

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

Careful: not × √2√2 ✗

Top times top, bottom times bottom.

= 2 × (1 + √2)(1 − √2)(1 + √2)

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: 1 squared minus (√2) squared.

√2 squared is 2.

So it is 1 minus 2: −1.

No root at the bottom now.

1² − (√2)² = 1 − 2 = −1

So the bottom is −1.

= 2 × (1 + √2)−1

3 Top: open the brackets

On top: 2 times every part of the bracket.

That is 2 + 2√2.

2 × (1 + √2) = 2 + 2√2

So we get (2 + 2√2) over −1.

= 2 + 2√2−1

4 Simplify

The bottom is negative.

We divide the top and the bottom by negative 1: every sign changes.

÷ (−1) top and bottom

Now the bottom is 1.

= −2 − 2√21

Over 1 is just the top: −2 − 2√2.

= −2 − 2√2

5 Check

Let's check with a calculator.

The start is about -4.83.

Start: 21 − √2 ≈ -4.83

The answer is about -4.83 too.

So it is right!

Answer: −2 − 2√2 ≈ -4.83

Let's look back.

First: find the partner.

Then: bottom: difference of squares.

Then: top: open the brackets.

Then: simplify.

Last: check.

Answer