Surds – SS2 Mathematics Lesson Note (Second Term)

Surds are irrational numbers: roots of rational numbers whose value cannot be expressed as exact fractions, such as √2, √7, √12 and √18. A surd can be simplified to its basic form or written as a single surd, and surds can be added, subtracted, multiplied, divided and rationalised using a few rules.

Behavioural objectives

By the end of the lesson, students should be able to:

• state the rules of surds;
• identify similar surds and the conjugate of a surd;
• simplify surds to their basic form and express them as a single surd;
• add, subtract, multiply and divide surds;
• rationalise the denominator of a fraction involving surds;
• find the square root of expressions of the form a + b√c using equality of surds.

What are surds?

Surds are irrational numbers. They are the roots of rational numbers whose value cannot be expressed as exact fractions. Examples of surds are √2, √7, √12, √18, etc.

What are the rules of surds?

1. √(a × b) = √a × √b
2. √(a / b) = √a / √b
3. √(a + b) ≠ √a + √b
4. √(a – b) ≠ √a – √b

What is the basic form of a surd?

√a is said to be in its basic form if a does not have a factor, other than 1, that is a perfect square. Examples are √6, √5, √3 and √2.

√18 is not in its basic form because it can be broken into √(9 × 2) = 3√2. Hence 3√2 is in its basic form.

What are similar surds?

Surds are similar if their irrational part contains the same number under the root sign, for example:

1. 3√n and 5√n
2. 6√2 and 7√2

What are conjugate surds?

Conjugate surds are two surds whose product is a rational number.

• The conjugate of √3 – √5 is √3 + √5.
• The conjugate of -2√7 + √3 is -2√7 – √3.
• In general, the conjugate of √x + √y is √x – √y.
• The conjugate of √x – √y is √x + √y.

How do we simplify surds?

Surds can be simplified either in the basic form or as a single surd.

Example 1

Simplify the following in its basic form: (a) √45 (b) √98

Solution

(a) √45 = √(9 × 5) = √9 × √5 = 3√5
(b) √98 = √(49 × 2) = √49 × √2 = 7√2

Example 2

Simplify the following as a single surd: (a) 2√5 (b) 17√2

Solution

(a) 2√5 = √4 × √5 = √(4 × 5) = √20
(b) 17√2 = √289 × √2 = √(289 × 2) = √578

How do we add and subtract surds?

Surds in their basic forms which are similar can be added or subtracted.

Example: Evaluate the following.

(a) √32 + 3√8 (b) 7√3 – √75 (c) 3√48 – √75 + 2√12

Solution

(a) √32 + 3√8
= √(16 × 2) + 3√(4 × 2)
= 4√2 + 6√2
= 10√2

(b) 7√3 – √75
= 7√3 – √(25 × 3)
= 7√3 – 5√3
= 2√3

(c) 3√48 – √75 + 2√12
= 3√(16 × 3) – √(25 × 3) + 2√(4 × 3)
= 12√3 – 5√3 + 4√3
= 11√3

How do we multiply and divide surds?

Example: Evaluate the following.

(a) √45 × √28 (b) √24 / √50

Solution

(a) √45 × √28
= √(9 × 5) × √(4 × 7)
= 3√5 × 2√7
= 3 × 2 × √(5 × 7)
= 6√35

(b) √24 / √50
= √(24 / 50)
= √(12 / 25)
= √12 / √25
= √(4 × 3) / 5
= 2√3 / 5

What is rationalisation of surds?

Rationalisation of surds means multiplying the numerator and denominator by the denominator or by the conjugate of the denominator, so that the denominator becomes a rational number.

Example: Evaluate the following.

(a) 6/√3 (b) 3/(√3 + √2)

Solution

(a) 6/√3
= (6 × √3) / (√3 × √3)
= 6√3 / 3
= 2√3

(b) 3/(√3 + √2)
= 3(√3 – √2) / [(√3 + √2)(√3 – √2)]
= (3√3 – 3√2) / [(√3)² – (√2)²]
= (3√3 – 3√2) / (3 – 2)
= (3√3 – 3√2) / 1
= 3(√3 – √2)

What is equality of surds?

Given two expressions p + √m and q + √n, if p + √m = q + √n, then

p – q = √n – √m

The left side is a rational number and the right side is irrational, so they can be equal only if both are equal to zero (0).

p – q = 0, so p = q, and √n – √m = 0, so √n = √m (that is, n = m).

Example

Find the square root of the following: (a) 7 + 2√10 (b) 14 – 4√6

Solution

(a) Let the square root of 7 + 2√10 be √m + √n.

(√m + √n)² = 7 + 2√10
m + 2√(mn) + n = 7 + 2√10

Comparing both sides:

m + n = 7 ………… (i)
2√(mn) = 2√10, so √(mn) = √10
Squaring both sides: mn = 10 ………… (ii)

From equation (i), m = 7 – n. Put this in (ii):

(7 – n)n = 10
7n – n² = 10
n² – 7n + 10 = 0
n² – 2n – 5n + 10 = 0
n(n – 2) – 5(n – 2) = 0
(n – 5)(n – 2) = 0
n = 5 or 2

When n = 2, m = 7 – 2 = 5.
When n = 5, m = 7 – 5 = 2.
So m = 5 or 2.

The square root of 7 + 2√10 is √5 + √2.

(b) Let the square root of 14 – 4√6 be √P – √Q.

(√P – √Q)² = 14 – 4√6
P – 2√(PQ) + Q = 14 – 4√6

P + Q = 14 ………… (i)
-2√(PQ) = -4√6
√(PQ) = 2√6
Squaring both sides: PQ = (2√6)² = 4 × 6 = 24 ………… (ii)

From equation (i), P = 14 – Q. Substitute for P in equation (ii):

(14 – Q)Q = 24
14Q – Q² = 24
Q² – 14Q + 24 = 0
Q² – 12Q – 2Q + 24 = 0
Q(Q – 12) – 2(Q – 12) = 0
(Q – 12)(Q – 2) = 0
Q = 12 or 2

When Q = 12, P = 14 – 12 = 2.
When Q = 2, P = 14 – 2 = 12.

(√12 – √2) = (2√3 – √2) and
(√2 – √12) = (√2 – 2√3)

Evaluation

1. Simplify √75 in its basic form.
2. Express 3√7 as a single surd.
3. Write down the conjugate of √5 + √2.
4. Evaluate 2√18 + √8.
5. Rationalise 10/√5.

Assignment

1. Expand (3√2 – 1)(3√2 + 1). (a) 16 (b) 20 (c) 17 (d) 24
2. Simplify √200 in its basic form. (a) 10√2 (b) 5√4 (c) 2√10 (d) 2√50
3. Simplify 9/√3. (a) 3√2 (b) 3√3 (c) 1/3 (d) 2√2
4. Express 3√5 as a single surd. (a) √40 (b) √55 (c) √45 (d) √35
5. Simplify √128 – 4√8. (a) 0 (b) 1 (c) 2 (d) 3
6. Express (3√2 – √3) / (2√3 – √2) in the form √m/√n, where m and n are whole numbers.
7. Express 1/(√5 + √3) in the form p√5 + q√3, where p and q are rational numbers.