Rationalise 5/(4 + √6) Step by Step

Step by step

The question

Question: rationalise 5/(4 + √6).

54 + √6

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 4 + √6 is 4 − √6.

partner: 4 + √6 → 4 − √6

So we multiply by 4 − √6/4 − √6.

4 − √6/4 − √6 is 1, and times 1 changes nothing.

= 54 + √6 × 4 − √64 − √6

Careful!

Times √6 alone does not work here.

4 + √6 times √6 still has a root in it.

Careful: not × √6√6 ✗

Top times top, bottom times bottom.

= 5 × (4 − √6)(4 + √6)(4 − √6)

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: 4 squared minus (√6) squared.

√6 squared is 6.

So it is 16 minus 6: 10.

No root at the bottom now.

4² − (√6)² = 16 − 6 = 10

So the bottom is 10.

= 5 × (4 − √6)10

3 Top: open the brackets

On top: 5 times every part of the bracket.

That is 20 − 5√6.

5 × (4 − √6) = 20 − 5√6

So we get (20 − 5√6)/10.

= 20 − 5√610

4 Simplify

20, 5 and 10 can all be divided by 5.

20, 5, 10: all ÷ 5

So we get (4 − √6)/2.

= 4 − √62

5 Check

Let's check with a calculator.

The start is about 0.78.

Start: 54 + √6 ≈ 0.78

The answer is about 0.78 too.

So it is right!

Answer: 4 − √62 ≈ 0.78

Let's look back.

First: find the partner.

Then: bottom: difference of squares.

Then: top: open the brackets.

Then: simplify.

Last: check.

Answer