Rationalise 4/(1 - √3) Step by Step

Step by step

The question

Question: rationalise 4/(1 − √3).

41 − √3

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 − √3 is 1 + √3.

partner: 1 − √3 → 1 + √3

So we multiply by 1 + √3/1 + √3.

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

= 41 − √3 × 1 + √31 + √3

Careful!

Times √3 alone does not work here.

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

Careful: not × √3√3 ✗

Top times top, bottom times bottom.

= 4 × (1 + √3)(1 − √3)(1 + √3)

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 (√3) squared.

√3 squared is 3.

So it is 1 minus 3: −2.

No root at the bottom now.

1² − (√3)² = 1 − 3 = −2

So the bottom is −2.

= 4 × (1 + √3)−2

3 Top: open the brackets

On top: 4 times every part of the bracket.

That is 4 + 4√3.

4 × (1 + √3) = 4 + 4√3

So we get (4 + 4√3) over −2.

= 4 + 4√3−2

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 2.

= −4 − 4√32

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

4, 4, 2: all ÷ 2

Over 1 is just the top: −2 − 2√3.

= −2 − 2√3

5 Check

Let's check with a calculator.

The start is about -5.46.

Start: 41 − √3 ≈ -5.46

The answer is about -5.46 too.

So it is right!

Answer: −2 − 2√3 ≈ -5.46

Let's look back.

First: find the partner.

Then: bottom: difference of squares.

Then: top: open the brackets.

Then: simplify.

Last: check.

Answer