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Basic 9 · Mathematics · SURDS

Simplifying and Approximating Surds

In this topic you will simplify surd expressions, rationalise denominators and approximate the square roots of non-perfect squares using calculators and tables.

By the end of these notes you should be able to:

  • Expand brackets containing surds
  • Rationalise the denominator of a fraction
  • Approximate square roots

What you already know

  • Rules of surds

Copy everything that has a blue line on its left into your notebook. The other parts are for reading.

Lesson 1

Strand
Number
Sub-strand
SURDS
Indicator
B9.1.2.4.3 – Simplify given surds

Introduction

The distributive property works with surds just as it does with numbers and letters.

1. Single Brackets

One bracket

Expand √2(3 + √2).

  1. √2 × 3 = 3√2.
  2. √2 × √2 = 2.

Answer: 3√2 + 2

With simplification

Expand √3(√12 + 5).

  1. √3 × √12 = √36 = 6.
  2. √3 × 5 = 5√3.

Answer: 6 + 5√3

2. Double Brackets

To multiply two brackets, multiply each term in the first by each term in the second. This is often remembered as FOIL: First, Outer, Inner, Last.

Double brackets

Expand (2 + √3)(4 + √3).

  1. First: 2 × 4 = 8.
  2. Outer: 2 × √3 = 2√3.
  3. Inner: √3 × 4 = 4√3.
  4. Last: √3 × √3 = 3.
  5. Collect: 8 + 3 + 6√3.

Answer: 11 + 6√3

3. Conjugates

Conjugate: The same two terms as a compound surd with the middle sign changed.

Conjugate pairs
SurdConjugateProduct
2 + √32 − √34 − 3 = 1
3 + √23 − √29 − 2 = 7
√5 − 1√5 + 15 − 1 = 4

Multiplying conjugates uses the identity (a + b)(a − b) = a² − b². The middle terms cancel, and the result has no surd. This is a very useful fact, and you will need it in the next lesson.

Students doing algebra in exercise books
Figure 1: Working on algebra Expand three pairs of conjugates.

4. Squares of Compound Surds

Take care with squares. (2 + √3)² is not 4 + 3. Write it as (2 + √3)(2 + √3). This gives 4 + 2√3 + 2√3 + 3 = 7 + 4√3. Learners often forget the middle term. The pattern is (a + b)² = a² + 2ab + b². Check by using a calculator: 2 + √3 ≈ 3.732, and its square is about 13.93. 7 + 4√3 ≈ 7 + 6.93 = 13.93. The two agree, and this confirms the result.

New words

expand
To multiply out brackets.
conjugate
A surd with its middle sign changed.
FOIL
First, outer, inner, last.
identity
An equation true for all values.

Summary

  • Multiply every term in the first bracket by every term in the second.
  • Conjugates give a result with no surd.
  • (a + b)² has a middle term.

Class activity: FOIL Practice

  1. Expand five pairs of brackets.
  2. Check using a calculator.
  3. Compare your answers.

You need: Number cards, Square-grid paper, Calculator

Exercises

Answer in your exercise book first, then check your answers.

  1. 1. √5 × √5 =

    • A. 5
    • B. 25
    • C. 10
    • D. √10
    Show answer
    Answer: 5Rule.

  2. 2. The conjugate of 4 + √7 is…

    • A. 4 − √7
    • B. 7 + √4
    • C. −4 + √7
    • D. 4 + √7
    Show answer
    Answer: 4 − √7Change the sign.

  3. 3. (3 + √2)(3 − √2) = 7.

    • A. True
    • B. False
    Show answer
    Answer: True9 − 2.

  4. 4. (a + b)(a − b) = a² − ______.

    Show answer
    Answer: b²Identity.

  5. 5. Expand √2(√8 + 1).

    Show answer
    Answer: 4 + √2√16 + √2.

  6. 6. Expand (1 + √5)².

    Show answer
    Answer: 6 + 2√51 + 2√5 + 5.

Lesson 2

Strand
Number
Sub-strand
SURDS
Indicator
B9.1.2.4.3 – Simplify given surds

Introduction

Fractions with surds on the bottom are hard to work with. Mathematicians always prefer a whole number there.

1. What Is Rationalising?

Rationalise the denominator: To change a fraction so that its denominator has no surd.

We multiply the top and bottom of a fraction by the same value, which is the same as multiplying by 1. So the value of the fraction does not change, only its form. For a simple surd denominator, multiply by that surd. For a compound denominator, multiply by the conjugate.

2. Simple Denominators

Simple surd

Rationalise 1/√2.

  1. Multiply the top and the bottom by √2.
  2. (1 × √2)/(√2 × √2).

Answer: √2/2

With simplification

Rationalise 6/√3.

  1. Multiply the top and the bottom by √3.
  2. 6√3/3.

Answer: 2√3

3. Compound Denominators

Using a conjugate

Rationalise 1/(2 + √3).

  1. Multiply the top and the bottom by 2 − √3.
  2. Top: 2 − √3.
  3. Bottom: 4 − 3 = 1.

Answer: 2 − √3

Larger example

Rationalise 4/(3 − √5).

  1. Multiply the top and the bottom by 3 + √5.
  2. Top: 4(3 + √5).
  3. Bottom: 9 − 5 = 4.

Answer: 3 + √5

A teacher explaining at the board
Figure 2: Teacher at the board Rationalise five fractions.

4. Why Rationalise?

Before calculators, dividing by 1.414 by hand was very difficult, but √2/2 = 1.414 ÷ 2 was easy. Today, rationalising still helps in simplifying expressions, comparing values and adding fractions with surds. For example, 1/√2 and √2/2 look different, but they are equal. It is expected in most examinations, so a final answer with a surd in the denominator may lose marks.

5. Combining Surd Fractions

Adding

Simplify 1/√2 + √2.

  1. 1/√2 = √2/2.
  2. √2 = 2√2/2.
  3. Add: (√2 + 2√2)/2.

Answer: 3√2/2

New words

rationalise
To remove a surd from a denominator.
denominator
The bottom of a fraction.
conjugate
A surd with its sign changed.
equivalent
Having the same value.

Summary

  • Multiply the top and bottom by the same value.
  • Use the surd for simple denominators, and the conjugate for compound ones.
  • Simplify the final answer.

Class activity: Rationalising Relay

  1. Each member does one step.
  2. Pass the paper on.
  3. Check the final answer with a calculator.

You need: Number cards, Square-grid paper, Calculator

Exercises

Answer in your exercise book first, then check your answers.

  1. 1. Rationalise 1/√3.

    • A. √3/3
    • B. 3/√3
    • C. √3
    • D. 1/3
    Show answer
    Answer: √3/3Multiply by √3.

  2. 2. The denominator of 1/(1 + √2) is rationalised using…

    • A. 1 − √2
    • B. 1 + √2
    • C. √2
    • D. 2
    Show answer
    Answer: 1 − √2Conjugate.

  3. 3. Multiplying the top and bottom by the same value keeps a fraction's value.

    • A. True
    • B. False
    Show answer
    Answer: True× 1.

  4. 4. 10/√5 = ______√5.

    Show answer
    Answer: 210√5/5.

  5. 5. Rationalise 3/√6.

    Show answer
    Answer: √6/23√6/6.

  6. 6. Rationalise 2/(√3 − 1).

    Show answer
    Answer: √3 + 1Bottom = 2.

Lesson 3

Strand
Number
Sub-strand
SURDS
Indicator
B9.1.2.4.4 – Approximate the square roots of non-perfect squares with calculators/tables

Introduction

A builder cannot cut wood to exactly √2 m. Real measurements need decimal values.

1. Using a Calculator

Common square roots
RootApproximation (3 d.p.)
√21.414
√31.732
√52.236
√72.646
√103.162

To find a square root on a calculator, press the root key, then the number and equals (some models work the other way round). Then round to the required accuracy. Learn √2 and √3 by heart, since they appear often. You can use these values to work out others: √12 = 2√3 ≈ 2 × 1.732 = 3.464.

2. Estimating Without a Calculator

Between two integers

Between which two whole numbers does √30 lie?

  1. 25 < 30 < 36.
  2. √25 = 5 and √36 = 6.

Answer: √30 lies between 5 and 6

Average method

Estimate √20.

  1. Guess 4.5.
  2. 20 ÷ 4.5 ≈ 4.44.
  3. Average: (4.5 + 4.44) ÷ 2 ≈ 4.47.

Answer: √20 ≈ 4.47

This method is called the divide-and-average method. Each round brings you closer to the true value. A calculator gives 4.472, so one round was already good.

3. Using Tables

Before calculators, people used printed tables of square roots. To use one, find the first digits of the number along the side and the next digit along the top. Then read off the value. Some schools still have such tables, and they are useful in a power cut. Rounding is required in both cases, since the true decimal never ends.

A builder measuring with a tape
Figure 3: Builder measuring Estimate three square roots, then check them.

4. Applications

Checking a square corner

A mason marks a wall corner 3 m along one wall and 4 m along the other. How long should the diagonal be? For a 2 m by 2 m corner?

  1. 3² + 4² = 25, so the diagonal = √25 = 5 m.
  2. 2² + 2² = 8, so the diagonal = √8.
  3. √8 = 2√2 ≈ 2 × 1.414.

Answer: 5 m and about 2.83 m

Masons use this check to make sure that corners are square. If the measurement is wrong, the corner is not a right angle. This is the Pythagoras theorem at work on a building site in Ghana.

New words

approximate
Close to but not exact.
estimate
A rough calculation.
divide-and-average
A method for finding square roots.
Pythagoras' theorem
a² + b² = c² in a right triangle.

Summary

  • Use calculators or tables and round the answer.
  • Estimate by finding the squares on either side.
  • Surds are used in real measurement.

Class activity: Corner Check

  1. Mark 3 m and 4 m along two walls.
  2. Measure the diagonal.
  3. Decide whether the corner is square.

You need: Calculators, Printed square-root reference table, Number cards

Exercises

Answer in your exercise book first, then check your answers.

  1. 1. √2 ≈

    • A. 1.414
    • B. 2.236
    • C. 1.732
    • D. 3.162
    Show answer
    Answer: 1.414Learn it.

  2. 2. √50 lies between…

    • A. 7 and 8
    • B. 6 and 7
    • C. 8 and 9
    • D. 5 and 6
    Show answer
    Answer: 7 and 849 and 64.

  3. 3. Decimal values of surds are approximations.

    • A. True
    • B. False
    Show answer
    Answer: TrueEndless.

  4. 4. √12 ≈ ______ (use √3 ≈ 1.732).

    Show answer
    Answer: 3.4642√3.

  5. 5. Estimate √10 using the average method, with first guess 3.

    Show answer
    Answer: 3.17 (10 ÷ 3 = 3.33, average 3.17).Average.

  6. 6. Find the diagonal of a 1 m square, to 2 d.p.

    Show answer
    Answer: 1.41 m√2.

Homework

  1. Measure a rectangular object, and calculate the diagonal to 2 d.p.
Simplifying and Approximating Surds – Basic 9 Mathematics Notes | sirbus