Worked examples
The question
Question: rationalise 1 over (√2 + 1).
1√2 + 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 √2 + 1 is √2 − 1.
partner: √2 + 1 → √2 − 1
So we multiply by √2 − 1 over √2 − 1.
√2 − 1 over √2 − 1 is 1, and times 1 changes nothing.
= 1√2 + 1 × √2 − 1√2 − 1
Careful!
Times √2 alone does not work here.
√2 + 1 times √2 still has a root in it.
Careful: not × √2√2 ✗
Top times top, bottom times bottom.
= 1 × (√2 − 1)(√2 + 1)(√2 − 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: (√2) squared minus 1 squared.
√2 squared is 2.
So it is 2 minus 1: 1.
No root at the bottom now.
(√2)² − 1² = 2 − 1 = 1
So the bottom is 1.
= 1 × (√2 − 1)1
3 Top: open the brackets
On top: 1 times every part of the bracket.
That is √2 − 1.
1 × (√2 − 1) = √2 − 1
So we get (√2 − 1)/1.
= √2 − 11
4 Simplify
Over 1 is just the top: √2 − 1.
= √2 − 1
5 Check
Let's check with a calculator.
The start is about 0.41.
Start: 1√2 + 1 ≈ 0.41
The answer is about 0.41 too.
So it is right!
Answer: √2 − 1 ≈ 0.41
Let's look back.
First: find the partner.
Then: bottom: difference of squares.
Then: top: open the brackets.
Then: simplify.
Last: check.
▶ Watch it solved
The question
Question: rationalise 3 over (√3 − 1).
3√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√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 × (√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 × (√3 + 1)2
3 Top: open the brackets
On top: 3 times every part of the bracket.
That is 3 + 3√3.
3 × (√3 + 1) = 3 + 3√3
So we get (3 + 3√3)/2.
The numbers can't be cancelled, so we are done.
= 3 + 3√32
4 Check
Let's check with a calculator.
The start is about 4.10.
Start: 3√3 − 1 ≈ 4.10
The answer is about 4.10 too.
So it is right!
Answer: 3 + 3√32 ≈ 4.10
Let's look back.
First: find the partner.
Then: bottom: difference of squares.
Then: top: open the brackets.
Last: check.
▶ Watch it solved
The question
Question: rationalise (√5 + 1) over (√5 − 1).
√5 + 1√5 − 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 √5 − 1 is √5 + 1.
partner: √5 − 1 → √5 + 1
So we multiply by √5 + 1 over √5 + 1.
√5 + 1 over √5 + 1 is 1, and times 1 changes nothing.
= √5 + 1√5 − 1 × √5 + 1√5 + 1
Careful!
Times √5 alone does not work here.
√5 − 1 times √5 still has a root in it.
Careful: not × √5√5 ✗
Top times top, bottom times bottom.
= (√5 + 1)(√5 + 1)(√5 − 1)(√5 + 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: (√5) squared minus 1 squared.
√5 squared is 5.
So it is 5 minus 1: 4.
No root at the bottom now.
(√5)² − 1² = 5 − 1 = 4
So the bottom is 4.
= (√5 + 1)(√5 + 1)4
3 Top: open the brackets
On top: every part times every part.
√5 times √5 is 5.
Then the like terms go together: 6 + 2√5.
1 + √5 + √5 + 5
= 6 + 2√5
So we get (6 + 2√5)/4.
= 6 + 2√54
4 Simplify
6, 2 and 4 can all be divided by 2.
6, 2, 4: all ÷ 2
So we get (3 + √5)/2.
= 3 + √52
5 Check
Let's check with a calculator.
The start is about 2.62.
Start: √5 + 1√5 − 1 ≈ 2.62
The answer is about 2.62 too.
So it is right!
Answer: 3 + √52 ≈ 2.62
Let's look back.
First: find the partner.
Then: bottom: difference of squares.
Then: top: open the brackets.
Then: simplify.
Last: check.
▶ Watch it solved
