Rationalising the denominator — single and conjugate — Mathematics

Before this

This leads to

Difference of two squares
Exact values and the unit circle · coming soon

The same number, written two ways

Here are two numbers: $\frac{1}{2}$ and $\frac{2}{2}$ . They look different. They are exactly equal. Both are about 0.7071.

Now imagine you have no calculator, and you know only that $2 \approx 1.41421356$ . To find the first number, you must divide 1 by 1.41421356. That is a long division by an eight-digit decimal, and it is slow and easy to get wrong. To find the second number, you halve 1.41421356. You can do that in your head: 0.70710678.

Same number. One form is hard to work with, and the other is easy. For hundreds of years, people calculated by hand, and that difference mattered. So mathematicians agreed on a habit: do not leave a surd in the denominator of a fraction. Rewriting a fraction so that its denominator is rational is called rationalising the denominator.

Calculators have made the division easy. The habit has survived for a better reason, which you will see soon: the rationalised form shows you which surds are like surds. Examiners also expect it. An answer such as $\frac{1}{2}$ is usually not accepted as "simplest form".

This lecture handles two kinds of denominator. The first, a single surd such as $2$ , needs only one idea. The second, a bracket such as $3 + 2$ , needs the conjugate pairs from the last lecture.

Multiplying by 1 in disguise

Here is the key fact. Any number divided by itself is 1, so $\frac{2}{2} = 1$ . Multiplying a fraction by 1 does not change its value. It only changes how the fraction is written.

So multiply $\frac{1}{2}$ by $\frac{2}{2}$ . The top becomes $1 \times 2 = 2$ . The bottom becomes $2 \times 2 = 2$ , because a square root times itself gives the number inside. That fact from the last lecture does all the work here.

\frac{1}{2} = \frac{1}{2} \times \frac{2}{2} = \frac{2}{2}

The denominator is now 2, a rational number. The value has not changed. Only the form has changed.

The same move works for any single surd. For $\frac{3}{5}$ , multiply top and bottom by $5$ :

\frac{3}{5} = \frac{3 \times 5}{5 \times 5} = \frac{3 5}{5}

When there is a number in front of the surd, as in $\frac{4}{3 7}$ , you only need to multiply by the surd part, $7$ . The 3 is already rational, so it can stay:

\frac{4}{3 7} = \frac{4 \times 7}{3 7 \times 7} = \frac{4 7}{3 \times 7} = \frac{4 7}{21}

Check with decimals: $4 \div (3 \times 2.6458) \approx 0.504$ , and $4 \times 2.6458 \div 21 \approx 0.504$ . Both forms give the same number, as they must.

Why the rationalised form is worth having

Try to simplify $\frac{1}{2} + 2$ . In this form, the two terms do not look alike. One has $2$ on the bottom, and the other has it on the top. Rationalise the first term, and the picture changes:

\frac{1}{2} + 2 = \frac{2}{2} + 2 = \frac{1}{2} 2 + 12 = \frac{3}{2} 2

Now both terms are multiples of $2$ . They are like surds, and in the lecture on adding surds you saw that like surds can be collected. Rationalising made the match visible. Check: $0.7071 + 1.4142 = 2.1213$ , and $1.5 \times 1.4142 = 2.1213$ .

When the denominator is a bracket

Now try $\frac{1}{3 + 2}$ . The obvious move is to multiply by $\frac{2}{2}$ again. Watch what happens to the bottom:

(3 + 2) \times 2 = 32 + 2

The surd has not gone. It has only moved from one term to the other. Multiplying by the surd alone works only when the surd is alone. Here it is joined to a 3, and the 3 picks up the $2$ .

We need a partner for $3 + 2$ that turns it into a whole number. Before you read on, make a guess below.

Predict — multiply by what?

You want to rationalise 1 ÷ (3 + √2). What should you multiply it by? Choose one, then read the explanation.

Four ways to try to rationalise 1 ÷ (3 + √2), and what each one does to the denominator:

Multiply by √2 ÷ √2: the bottom becomes 3√2 + 2. The surd is still there.
Multiply by (3 + √2) ÷ (3 + √2): the bottom becomes 11 + 6√2. The surd is still there.
Multiply by (3 − √2) ÷ (3 − √2): the bottom becomes 9 − 2 = 7. This works.
Multiply only the bottom by (3 − √2): the bottom becomes 7, but the fraction is now 1 ÷ 7 ≈ 0.143 instead of ≈ 0.226. Its value has changed, so this is wrong.

The last lecture promised that conjugate pairs would remove surds from the bottom of a fraction. Here is that promise kept. The conjugate of $3 + 2$ is $3 - 2$ : the same two terms with the sign between them changed. Their product has no surd:

(3 + 2) (3 - 2) = 9 - 32 + 32 - 2 = 7

So multiply top and bottom by the conjugate. The top becomes $3 - 2$ , and the bottom becomes 7:

\frac{1}{3 + 2} = \frac{1}{3 + 2} \times \frac{3 - 2}{3 - 2} = \frac{3 - 2}{7}

Check: $1 \div 4.4142 \approx 0.2265$ , and $(3 - 1.4142) \div 7 \approx 0.2265$ .

Rationalising the denominator

If the denominator is $k a$ , multiply top and bottom by $a$ . The bottom becomes $ka$ .

If the denominator is $p + b$ or $p - b$ , multiply top and bottom by its conjugate, $p - b$ or $p + b$ . The bottom becomes $p^{2} - b$ .

In both cases you multiply by a fraction equal to 1, so the value never changes.

Why the conjugate always works

The conjugate is not a lucky trick. It works every time, and the reason is worth seeing.

When you multiply $p + b$ by $p - b$ , the two middle terms are $- p b$ and $+ p b$ . They are equal and opposite, so they cancel. What is left is $p \times p$ and $b \times (- b)$ . The surd is multiplied by itself, and a square root times itself is the number inside. So the product is $p^{2} - b$ , with no surd left.

That is it. That is the whole idea: the conjugate forces the surd to be squared, and squaring removes a square root.

There is also a picture. The area $p^{2} - b$ is a square of side $p$ with a smaller square of side $b$ cut from its corner. Cut the L-shape that is left into two pieces, and they fit together as a rectangle with sides $p + b$ and $p - b$ .

With p = 3 and √b = √2. Top: the 3 × 3 square with a √2 × √2 square cut from its corner has area 9 − 2 = 7. Bottom: piece B moves to the end of piece A. The rectangle has sides 3 + √2 and 3 − √2, and its area is still 7, a whole number.

You will meet this pattern again in algebra as the difference of two squares, $(x + y) (x - y) = x^{2} - y^{2}$ . Rationalising with a conjugate is that identity with $y = b$ .

Stop and think

A student rationalises $\frac{1}{3 + 2}$ like this: "The bottom times $3 - 2$ is 7, so the answer is $\frac{1}{7}$ ." What went wrong?

Show the answer

The student multiplied only the bottom. That is multiplying the whole fraction by $\frac{1}{3 - 2}$ , which is not 1. So the value changed: $\frac{1}{7} \approx 0.143$ , but the original is $\approx 0.226$ . The method only works because $\frac{3 - 2}{3 - 2} = 1$ . The top must be multiplied too, giving $\frac{3 - 2}{7}$ .

The hard part is the top, not the bottom

When the top of the fraction is also a bracket, you must expand two brackets on top. The bottom is easy: it is always $p^{2} - b$ . The top is where most mistakes happen, especially with minus signs. Write all four products before you collect anything.

The working below has exactly one wrong line. Find it.

Spot the error — rationalising (4 − √3) ÷ (2 + √3)

One line contains a mistake. Tap the line you think is wrong. Every line has an explanation.

The working, with the mistake marked and corrected:

Line 1. Multiply top and bottom by the conjugate of 2 + √3, which is 2 − √3. (Correct.)
Line 2. The top is (4 − √3)(2 − √3), with four products. (Correct.)
Line 3. 4 × 2 = 8, 4 × (−√3) = −4√3, and (−√3) × 2 = −2√3. (Correct.)
Line 4. (−√3) × (−√3) = −3. The mistake: a negative times a negative is positive, so this is +3.
Line 5. The top is 8 − 4√3 − 2√3 − 3 = 5 − 6√3. (This carries the mistake. It should be 8 − 4√3 − 2√3 + 3 = 11 − 6√3.)
Line 6. The bottom is (2 + √3)(2 − √3) = 4 − 3 = 1. (Correct.)
Line 7. The answer is 5 − 6√3. (It should be 11 − 6√3 ≈ 0.608.)

Notice how the decimal check caught the mistake at once. The wrong answer was negative, but the original fraction is clearly positive. A ten-second check with rough decimals is the best habit you can build for this topic.

Worked examples

Worked Example 1 — A single surd

Rationalise $\frac{5}{3}$ .

Multiply top and bottom by $3$ :

\frac{5}{3} = \frac{5 3}{3 \times 3} = \frac{5 3}{3}

Check: $5 \div 1.7321 \approx 2.887$ , and $5 \times 1.7321 \div 3 \approx 2.887$ .

Worked Example 2 — Simplify after you rationalise

Rationalise $\frac{6}{2 5}$ .

Multiply top and bottom by $5$ . The bottom becomes $2 \times 5 = 10$ :

\frac{6}{2 5} = \frac{6 5}{10} = \frac{3 5}{5}

The last step cancels the common factor 2 from 6 and 10. Always look for this after rationalising. Check: $6 \div 4.4721 \approx 1.342$ , and $3 \times 2.2361 \div 5 \approx 1.342$ .

Worked Example 3 — The fraction disappears

Rationalise $\frac{1}{5 - 2}$ .

The conjugate of $5 - 2$ is $5 + 2$ . The bottom becomes $(5)^{2} - 2^{2} = 5 - 4 = 1$ :

\frac{1}{5 - 2} = \frac{5 + 2}{1} = 5 + 2

The denominator became 1, so there is no fraction left at all. Check: $1 \div 0.2361 \approx 4.236$ , and $2.2361 + 2 = 4.2361$ .

Worked Example 4 — A bracket on top as well

Rationalise $\frac{1 + 3}{2 - 3}$ .

Multiply top and bottom by the conjugate, $2 + 3$ . The bottom is $4 - 3 = 1$ . The top has four products:

(1 + 3) (2 + 3) = 2 + 3 + 23 + 3 = 5 + 33

So the answer is $5 + 33$ . Check: $2.7321 \div 0.2679 \approx 10.20$ , and $5 + 3 \times 1.7321 \approx 10.20$ .

Worked Example 5 — A negative denominator

Rationalise $\frac{3 + 2}{1 + 2}$ .

Multiply top and bottom by $1 - 2$ . The bottom is $1 - 2 = - 1$ . The top is:

(3 + 2) (1 - 2) = 3 - 32 + 2 - 2 = 1 - 22

Dividing by $- 1$ changes the sign of every term on top:

\frac{1 - 2 2}{- 1} = - 1 + 22 = 22 - 1

A denominator of $p^{2} - b$ is negative whenever $b$ is bigger than $p^{2}$ . That is fine, as long as the sign goes to every term. Check: $4.4142 \div 2.4142 \approx 1.828$ , and $2 \times 1.4142 - 1 \approx 1.828$ .

Three classic mistakes

Trap 1 — Multiplying only the bottom

The method works because you multiply by a fraction equal to 1. If only the bottom is multiplied, the value of the fraction changes. Whatever multiplies the bottom must multiply the top.

Trap 2 — Keeping the same sign

Multiplying $3 + 2$ by $3 + 2$ squares the bracket and gives $11 + 62$ . The surd survives. The conjugate has the opposite sign in the middle, so that the two middle terms cancel.

Trap 3 — Cancelling only part of the top

$\frac{4 + 2 3}{2}$ is $2 + 3$ , because both terms on top are divided by 2. Writing $2 + 23$ divides only the first term. A fraction bar divides the whole of the top.

Try it yourself

Work each one on paper first. Open a hint only when you are stuck.

Rationalise and simplify $\frac{10}{5}$ and $\frac{3}{2 6}$ .

Hint 1
Multiply top and bottom by the surd on the bottom: $5$ for the first, $6$ for the second.

Hint 2
After rationalising, look for a common factor between the number on top and the number on the bottom.

Full answer
$\frac{10}{5} = \frac{10 5}{5} = 25$ . $\frac{3}{2 6} = \frac{3 6}{12} = \frac{6}{4}$ . Check: $10 \div 2.2361 \approx 4.472$ and $2 \times 2.2361 \approx 4.472$ ; $3 \div 4.899 \approx 0.612$ and $2.449 \div 4 \approx 0.612$ .
Rationalise $\frac{2}{3 - 5}$ .

Hint 1
The conjugate of $3 - 5$ is $3 + 5$ .

Hint 2
The bottom becomes $9 - 5 = 4$ . Then look for a common factor of 2.

Full answer
$\frac{2 ( 3 + 5 )}{9 - 5} = \frac{2 ( 3 + 5 )}{4} = \frac{3 + 5}{2}$ . Check: $2 \div 0.7639 \approx 2.618$ , and $(3 + 2.2361) \div 2 \approx 2.618$ .
Rationalise $\frac{3 + 1}{3 - 1}$ , and give your answer in the form $a + b$ .

Hint 1
Multiply top and bottom by $3 + 1$ . The top becomes $(3 + 1)^{2}$ .

Hint 2
$(3 + 1)^{2} = 3 + 23 + 1$ , and the bottom is $3 - 1 = 2$ .

Full answer
Top: $(3 + 1)^{2} = 4 + 23$ . Bottom: $3 - 1 = 2$ . So the fraction is $\frac{4 + 2 3}{2} = 2 + 3$ , dividing both terms on top by 2. Check: $2.7321 \div 0.7321 \approx 3.732$ , and $2 + 1.7321 = 3.7321$ .
Show that $\frac{1}{2 + 1} + \frac{1}{3 + 2} + \frac{1}{4 + 3} = 1$ exactly.

Hint 1
Rationalise each fraction on its own. The conjugate of $3 + 2$ is $3 - 2$ , and the same method works with two surds.

Hint 2
Each denominator becomes 1. For example, $(3 + 2) (3 - 2) = 3 - 2 = 1$ . Then add the three answers and look for terms that cancel.

Full answer
Each bottom becomes 1: $2 - 1 = 1$ , $3 - 2 = 1$ and $4 - 3 = 1$ . So the three fractions are $2 - 1$ , $3 - 2$ and $4 - 3$ . Adding them, $2$ and $3$ each appear once with a plus and once with a minus, so they cancel. What is left is $4 - 1 = 2 - 1 = 1$ . Check: $0.4142 + 0.3178 + 0.2679 \approx 1.000$ . Three irrational fractions add to exactly 1, and rationalising is what makes this visible.

The bigger idea

Rationalising is a small example of a big habit in mathematics: when a form is awkward, multiply by 1 in a clever disguise. The value stays the same, and the form becomes useful. The disguise here is the conjugate, and it works because squaring removes a square root.

This is also where surds pay off. Exact answers such as $\frac{3}{2}$ and $\frac{2}{2}$ appear throughout trigonometry, and the quadratic formula produces answers in conjugate pairs. Over this topic, you have gone from asking what a surd is to handling it as easily as a fraction.

Key concepts in this lesson

Rationalising the denominator means rewriting a fraction so that there is no surd in its denominator, without changing its value.
For a denominator such as $k a$ , multiplying top and bottom by $a$ works because $a \times a = a$ .
For a denominator such as $p + b$ , multiplying top and bottom by the conjugate $p - b$ gives a denominator of $p^{2} - b$ , because the middle terms cancel.
The method multiplies by a fraction equal to 1, so the top and the bottom must always be multiplied by the same thing.
After rationalising, simplify any common factor, divide every term on top, and check the answer with decimals.