Rationalise (2 + √2)/(2 - √2) Step by Step

Step by step

The question

Question: rationalise (2 + √2)/(2 − √2).

2 + √22 − √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 2 − √2 is 2 + √2.

partner: 2 − √2 → 2 + √2

So we multiply by 2 + √2/2 + √2.

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

= 2 + √22 − √2 × 2 + √22 + √2

Careful!

Times √2 alone does not work here.

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

Careful: not × √2√2 ✗

Top times top, bottom times bottom.

= (2 + √2)(2 + √2)(2 − √2)(2 + √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: 2 squared minus (√2) squared.

√2 squared is 2.

So it is 4 minus 2: 2.

No root at the bottom now.

2² − (√2)² = 4 − 2 = 2

So the bottom is 2.

= (2 + √2)(2 + √2)2

3 Top: open the brackets

On top: every part times every part.

√2 times √2 is 2.

Then the like terms go together: 6 + 4√2.

4 + 2√2 + 2√2 + 2

= 6 + 4√2

So we get (6 + 4√2)/2.

= 6 + 4√22

4 Simplify

6, 4 and 2 can all be divided by 2.

6, 4, 2: all ÷ 2

Over 1 is just the top: 3 + 2√2.

= 3 + 2√2

5 Check

Let's check with a calculator.

The start is about 5.83.

Start: 2 + √22 − √2 ≈ 5.83

The answer is about 5.83 too.

So it is right!

Answer: 3 + 2√2 ≈ 5.83

Let's look back.

First: find the partner.

Then: bottom: difference of squares.

Then: top: open the brackets.

Then: simplify.

Last: check.

Answer