Year 10 Standard · Unit 1 Number sets, surds and accuracy · Lesson 5 of 6

How do you move a surd out of the denominator?

A fraction with a surd on the bottom is hard to estimate and hard to combine. This lesson rewrites it with a whole-number denominator, and uses a square's diagonal to test whether the new form is really the same number.

Number and AlgebraNo calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. From Lesson 4: simplify √28.
    Answer2√7, because 28 = 4 × 7 and √4 = 2.
  2. From Year 8: expand 3(x + 4).
    Answer3x + 12. Multiply each term in the bracket by 3.
  3. From Year 9: expand (x + 3)(x − 3).
    Answerx² − 9. The middle terms, +3x and −3x, cancel.
Words used in this lesson
rationalize
rewrite a fraction so that its denominator has no surd. For example: 2/√7 = 2√7/7.
denominator
the bottom number of a fraction. For example: √5 in 1/√5.
numerator
the top number of a fraction. For example: 10 in 10/√5.
expand
multiply out brackets. For example: √2(√2 + 3) = 2 + 3√2.
difference of two squares
a² − b², which factorizes as (a + b)(a − b). For example: (4 + √3)(4 − √3) = 16 − 3.
exact value
a value written without rounding. For example: √7 rather than 2.65.

A tile measured two ways

A tiler in Marrakesh needs square tiles whose diagonal is exactly 10 cm. One plan gives the side as 10/√2 cm; another gives it as 5√2 cm.

Predict first: are these the same length? Which one could you estimate in your head? Then use the first figure, which finds the side of a square from its diagonal in two ways.

2
√2side, rationalized
2/√2side, as diagonal ÷ √2

Diagonal 2: the half diagonals are 1, at right angles, so side² = 1² + 1² = 2 and the side is √2 ≈ 1.41. The side is also 2/√2, so 2/√2 = √2.

Show me the steps
  1. Diagonal 2: the half diagonals are 1 and 1, so side² = 2 and the side is √2. The side is also 2/√2, so 2/√2 = √2.
  2. Diagonal 6: the half diagonals are 3, so side² = 9 + 9 = 18 and the side is √18 = 3√2 ≈ 4.24. So 6/√2 = 3√2.
  3. Diagonal 1: the half diagonals are 0.5, so side² = 0.25 + 0.25 = 0.5 and the side is √2/2 ≈ 0.71. So 1/√2 = √2/2.
  4. Diagonal 5: the side is 5√2/2 ≈ 3.54, the same length as 5/√2. Multiplying the top and bottom of 5/√2 by √2 gives 5√2/2 directly.

To rationalize a/√b, multiply the top and the bottom by √b. The bottom becomes √b × √b = b, a whole number, and because √b/√b = 1 the value does not change:

a/√b = a√b/b, then simplify. For a/(k√b), the bottom becomes k√b × √b = kb.

Brackets with surds expand like algebra, using √b × √b = b. In particular (a + √b)(a − √b) = a² − b, a whole number, because the middle terms cancel.

1
2
√2/2rationalized
0.71value, to 2 decimal places

1/√2 × √2/√2 = √2/2. Both are about 0.71: multiplying by √2/√2 multiplies by 1, so the number does not move.

Show me the steps
  1. a = 1, b = 2: 1/√2 × √2/√2 = √2/2. Both forms sit at about 0.71.
  2. a = 3, b = 6: 3/√6 × √6/√6 = 3√6/6, and 3 and 6 share the factor 3, so it simplifies to √6/2 ≈ 1.22.
  3. a = 5, b = 5: 5/√5 = 5√5/5 = √5 ≈ 2.24. A number divided by its own root is that root.
  4. b = 4: √4 = 2 is already a whole number, so there is nothing to rationalize: 3/√4 = 3/2.

Two common mistakes. Multiplying only the top or only the bottom by √b, which changes the value: both must be multiplied. And writing √b × √b = b² (or √7 × √7 = 49): a root times itself is the number under the root, so √7 × √7 = 7.

Worked examples

1. Rationalize 14/√7.

  1. Multiply the top and the bottom by √7: (14 × √7)/(√7 × √7) = 14√7/7.
  2. 14 ÷ 7 = 2, so 14/√7 = 2√7.

2. Expand and simplify (2 + √3)(1 + √3).

  1. Multiply every term by every term: 2 × 1 + 2 × √3 + √3 × 1 + √3 × √3.
  2. That is 2 + 2√3 + √3 + 3.
  3. Collect: 5 + 3√3.

Your go. The first step is given. Rationalize 9/√3.

  1. Multiply the top and the bottom by √3: (9 × √3)/(√3 × √3).
  2. Now work out the bottom and simplify the fraction.
Check the answer

√3 × √3 = 3, so it is 9√3/3 = 3√3.

Check before you go on

1. Which of these is 1/√11 written with a whole-number denominator?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. √3(√3 + 5) can be written as a + 5√3. What is a?

3. Rationalize 10/√5. It can be written as ?√5. What number goes in front of √5?

Level 2 · Building

4. Expand and simplify (3 + √2)(3 − √2). The answer is a whole number.

5. Rationalize 15/(2√5). It can be written as ?√5. What number goes in front of √5? Give it as a decimal.

6. A straight path crosses a rectangular park in Seoul from one corner to the opposite corner. The park is 30 m by 90 m. The path is ?√10 m long. What number goes in front of √10?

Level 3 · Stretch

7. A rectangle has area 24 cm² and width √6 cm. Its length can be written as ?√6 cm. What number goes in front of √6?

8. Expand (2 + √5)². It can be written as a + 4√5. What is a?

Something to think about

Rationalize 1/(√2 − 1). Multiplying by √2/√2 does not work here. What could you multiply the top and bottom by instead?

One way to think about it

Multiply by (√2 + 1)/(√2 + 1). The bottom becomes (√2 − 1)(√2 + 1) = 2 − 1 = 1, because the middle terms cancel, just as in (x − 1)(x + 1) = x² − 1. The top is √2 + 1, so 1/(√2 − 1) = √2 + 1, about 2.41.

Exit ticket

9. Which of these is equal to 12/√3?