Rationalise (√3 + 1)/(√3

Step by step

The question

Question: rationalise (√3 + 1) over (√3 − 1).

√3 + 1√3 − 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 √3 − 1 is √3 + 1.

partner: √3 − 1 → √3 + 1

So we multiply by √3 + 1 over √3 + 1.

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

= √3 + 1√3 − 1 × √3 + 1√3 + 1

Careful!

Times √3 alone does not work here.

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

Careful: not × √3√3 ✗

Top times top, bottom times bottom.

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

√3 squared is 3.

So it is 3 minus 1: 2.

No root at the bottom now.

(√3)² − 1² = 3 − 1 = 2

So the bottom is 2.

= (√3 + 1)(√3 + 1)2

3 Top: open the brackets

On top: every part times every part.

√3 times √3 is 3.

Then the like terms go together: 4 + 2√3.

1 + √3 + √3 + 3

= 4 + 2√3

So we get (4 + 2√3)/2.

= 4 + 2√32

4 Simplify

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

4, 2, 2: all ÷ 2

Over 1 is just the top: 2 + √3.

= 2 + √3

5 Check

Let's check with a calculator.

The start is about 3.73.

Start: √3 + 1√3 − 1 ≈ 3.73

The answer is about 3.73 too.

So it is right!

Answer: 2 + √3 ≈ 3.73

Let's look back.

First: find the partner.

Then: bottom: difference of squares.

Then: top: open the brackets.

Then: simplify.

Last: check.

Answer