Rationalise 6/(3 - √3) Step by Step

Step by step

The question

Question: rationalise 6/(3 − √3).

63 − √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 3 − √3 is 3 + √3.

partner: 3 − √3 → 3 + √3

So we multiply by 3 + √3/3 + √3.

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

= 63 − √3 × 3 + √33 + √3

Careful!

Times √3 alone does not work here.

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

Careful: not × √3√3 ✗

Top times top, bottom times bottom.

= 6 × (3 + √3)(3 − √3)(3 + √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: 3 squared minus (√3) squared.

√3 squared is 3.

So it is 9 minus 3: 6.

No root at the bottom now.

3² − (√3)² = 9 − 3 = 6

So the bottom is 6.

= 6 × (3 + √3)6

3 Top: open the brackets

On top: 6 times every part of the bracket.

That is 18 + 6√3.

6 × (3 + √3) = 18 + 6√3

So we get (18 + 6√3)/6.

= 18 + 6√36

4 Simplify

18, 6 and 6 can all be divided by 6.

18, 6, 6: all ÷ 6

Over 1 is just the top: 3 + √3.

= 3 + √3

5 Check

Let's check with a calculator.

The start is about 4.73.

Start: 63 − √3 ≈ 4.73

The answer is about 4.73 too.

So it is right!

Answer: 3 + √3 ≈ 4.73

Let's look back.

First: find the partner.

Then: bottom: difference of squares.

Then: top: open the brackets.

Then: simplify.

Last: check.

Answer