Rationalise (√6 + 2)/(√6

Step by step

The question

Question: rationalise (√6 + 2) over (√6 − 1).

√6 + 2√6 − 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 √6 − 1 is √6 + 1.

partner: √6 − 1 → √6 + 1

So we multiply by √6 + 1 over √6 + 1.

√6 + 1 over √6 + 1 is 1, and times 1 changes nothing.

= √6 + 2√6 − 1 × √6 + 1√6 + 1

Careful!

Times √6 alone does not work here.

√6 − 1 times √6 still has a root in it.

Careful: not × √6√6 ✗

Top times top, bottom times bottom.

= (√6 + 2)(√6 + 1)(√6 − 1)(√6 + 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: (√6) squared minus 1 squared.

√6 squared is 6.

So it is 6 minus 1: 5.

No root at the bottom now.

(√6)² − 1² = 6 − 1 = 5

So the bottom is 5.

= (√6 + 2)(√6 + 1)5

3 Top: open the brackets

On top: every part times every part.

√6 times √6 is 6.

Then the like terms go together: 8 + 3√6.

2 + 2√6 + √6 + 6

= 8 + 3√6

So we get (8 + 3√6)/5.

The numbers can't be cancelled, so we are done.

= 8 + 3√65

4 Check

Let's check with a calculator.

The start is about 3.07.

Start: √6 + 2√6 − 1 ≈ 3.07

The answer is about 3.07 too.

So it is right!

Answer: 8 + 3√65 ≈ 3.07

Let's look back.

First: find the partner.

Then: bottom: difference of squares.

Then: top: open the brackets.

Last: check.

Answer