Topic 12: Numerical Concept 2 (Surds)

By the end of this topic, you will be able to:

(a) use surds to represent roots that cannot be represented exactly as decimals.

(b) manipulate and simply expressions with surds.

Keywords

• denominator
• numerator
• radical expression
• radical number
• rationalising
• surds

Introduction

In this topic, you will study how to manipulate quantities, expressions and numerical concepts without exact square roots so as to be able to express such lengths more appropriately.

You have learnt Pythagoras’ theorem in Topic 10 and used it to find the length of the unknown side of a right-angled triangle. For some right-angled triangles, the theorem did not give you exact values, the lengths of the unknown sides were not exact. E

xpressing such results as decimals would result in writing long numbers and rounding off would create errors in accuracy. Most engineers prefer writing calculated values in the format of Va. For example, √2 instead of writing 1.414213562…

For example, look at the structure in the picture on the previous page. Some distances, e.g. gaps between successive poles, may not be exact. These may be written more appropriately in the form Va for convenience and accuracy.

ICT Support: Watch a short video on the address below: https://www.google.com/url?sa=t&source=web&rct=j&url=https://m. youtube.com/watch?v=568dGLFTom8&ved= 2ahUKEwjL346x8q3xAhVDXROKHTI_Ay0Q07QBegQIBhAE&usg= AOvVaw0k0eEUvc-WkPr0gPzK2Lt_

12.1 Identifying surds

From the short video, you should have learnt that va is a surd if the result is not exact. Thus √2, √3, √5, √7 e.t.c. are all examples of surds since their results are not exact. That is, they give rise to irrational numbers. But va is not a surd if the result is rational (or exact).

12.1.1 Identifying fundamental rules of surds

You have already identified that surds do not give rise to an exact value when computed. In some of the questions in Exercise 12.1, you may have come up with answers that are big and with a composite number under the root sign. Such big numbers may be broken down to simpler ones. In this section, you will establish some fundamental rules that will be used to break down and simplify such big surds. Now, take a study through the following presentation.

12.1.2 Finding the conjugate of a surd

In the previous section, you learnt that va x vb = √a x b. In this section, we shall look at the conjugate of a surd.

Activity 12.1 Finding conjugate of a simple surd

1. Investigate the results of the following operations.

(a) √2 × √2

(b) √3 x √3

(c) √5 x √5

2. What do you notice in (1)?

3. Share your findings with the class.

Activity 12.2 Finding conjugate of a compound surd

In Topic 6, you studied that:

(a + b) (c + d) = = ac + ad + bc + bd

a2-b2 = (a – b)(a + b)

1. Use this knowledge to expand:

(2 + √3)(2 – √3)

2. What do you notice about the result in (1)?

3. Share your findings with the class. ICT Support Watch a short video on the link below. https://www.google.com/search?q=conjugate+surds&client =ms-android-ragentek&prmd=ivn&source=lnms&tbm=vid& sa=X&ved=2ahUKEwi-rM6S8LLxAhUDYYUKHVDqBFIQ_ AUoAnoECAIQAg&biw=360&bih=592&dpr= 2#fpstate=ive&vid=cid:1be3a4e4,vid:P9bduhzdN6g,st:0

Learning points

The product of a surd and its conjugate gives a rational number. The conjugate of a single surd va is va. The conjugate of a compound surd a + b/c is a – bvc.

12.2 Manipulating and simplifying expressions with surds

You have already learnt that a surd involves a root that is not exact. In this section, you will learn how to carry out different operations involving surds. You are also expected to simplify expressions involving surds, especially where the surd is “bulky” and can be broken down into smaller numbers.

12.2.1 Simplifying surds

In section 12.2, you noted that √64 = √4 x 16 = √4 x √16. Here, 4 and 16 are factors of 64.

In a similar way, look at 45. The factors of 45 include: (1 x 45), (3 x 15) and (5 x 9). Of all these factors, only 9 has an exact square root.

Thus, √45 = √5 x 9 = √5 x √9; But √9 = 3

Hence, you can write:

√45 = √5 × √9

= √5 x 3

= 3√5

The most consistent way to do this is to find the prime factors of 45 as follows:

45 = 3 x 3 x 5

= (3 × 3) × 5

= 32 x 5

Hence, √45 = √32 x 5

= 3√5

Now, consider √96

96 =2x2x2x2x2x3

=(2×2) (2 x 2) x 2 x 3

= (2 × 2)2 × 2 × 3

= √42 x 6

Thus √96 = √42 x 6 = 4√6

Learning point

“Bulk” surds can be broken down into “lighter” ones by considering two of its factors, one of which has a direct root.

Exercise 12.2

1. Write each of the following surds as complete square roots (i.e. in the form Va).

(a) 10√2

(b) 5√6 2.

(c) √27

(d) √54

(e) 7√3

(f) 3√11

2. Express the following in the simplest possible form of avb.

(b) √96

(c) √250

(d)√125

1. 2. 12.2.2 Adding and subtracting surds

Only surds with the same irrational factor can be added or subtracted. Examples

1. Add: (13+7√3) + (2 – 4√3)

Solution You notice that these two are surds involving the rational and irrational parts. So they are compound surds. The addition of such surds involves adding the rational parts together and then the irrational parts together. Thus:

(13+7√3) + (2-4√3) = (13+2)+(7√3 – 4√3)

= (15) + (3√3)

= 15+ 3√3

2. Simplify:

(a) 5√6-3√6

(b) 2√3 +3√2 – √3

Solution

(a) 5√6-3√6= (5-3)√6 = 2√6

(b) 2√3 + 3√2-√3 = 2√3-√3 + 3√2 t

= √3 +3√2

Learning point

Only similar surds can be added or subtracted.

Exercise 12.3

1. Simplify each of the following.

(a) 3√5 +5√5

(b) 2√6-4√6

(c) 3√3 + 4√3 +5√3

(d) 3√2-5√3-2√2 + 2√3

2. Simplify each of the following.

(a) √5+ √20

(b) √8+ √8

(c) √75+ √27

(d) √45 + √125

(e) √100-√1210 + √1440

(f) 2√18+ 4√72 + √50 + 3√98

12.2.3 Multiplying and dividing surds

You have already learnt that √a x vb = √a x b and that

This means that:

√a x √a = √a x a

= √a2

= a

Examples

1. Evaluate each of the following.

(a) √7 x √7

(b) 2√5 x 3√5

(c) 3√2 × √8 Va = √b

12.2.4 Rationalising a surd

You have already learnt that the product of a surd and its conjugate results in a rational number. In this section, you will use that idea and the idea of “difference of two squares” you learnt in Topic 6 to rationalise the denominator of given surds.