Mathematics Form 1-4 : CHAPTER FOURTY FOUR – SURDS

Specific Objectives

By the end of the topic the learner should be able to:

(a) Define rational and irrational numbers,

(b) Simplify expressions with surds;

(c) Rationalize denominators with surds.

Content

(a) Rational and irrational numbers

(b) Simplification of surds

(c) Rationalization of denominators.

Rational and irrational numbers

Rational numbers

A rational number is a number which can be written in the form , where p and q are integers and q.The integer’s p and q must not have common factors other than 1.

Numbers such as 2, are examples of rational numbers .Recurring numbers are also rational numbers.

Irrational numbers

Numbers that cannot be written in the form .Numbers such as are irrational numbers.

Surds

Numbers which have got no exact square roots or cube root are called surds e.g. ,,

The product of a surd and a rational number is called a mixed surd. Examples are

, and

Order of surds

,

,

Simplification of surds

A surd can be reduced to its lowest term possible, as follows

Example

Simplify

Solution

Operation of surds

Surds can be added or subtracted only if they are like surds (that is, if they have the same value under the root sign).

Example 1

Simplify the following.

1. 3 √2 + 5√2
2. 8 √5 − 2√5

Solution

1. 3 √2 + 5√2 = 8 √2
2. 8 √5 − 2√5 = 6√5

Summary

Let a =

Therefore = a + a

=2 a

But a =

Hence =

Multiplication and Division of surds

Surds of the same order can be multiplied or divided irrespective of the number under the root sign.

Law 1: √a x √b = √ab When multiplying surds together, multiply their values together.

e.g.1 √3 x √12 = √ (3 x 12) = √36 = 6

e.g.2 √7 x √5 = √35

This law can be used in reverse to simplify expressions…

e.g.3 √12 = √2 x √6 or √4 x √3 = 2√3

Law 2:√a ÷ √b or = √(a/b) When dividing surds, divide their values (and vice versa).

e.g.1 √12 = √(12 ÷ 3) = √4 = 2

√3

e.g.2

Law 3: √ (a²) or (√a) ² = a When squaring a square-root, (or vice versa), the symbols cancel

Each other out, leaving just the base.

e.g.1 √12² = 12

e.g.2 √7 x √7 = √7² = 7

Note:

 If you add the same surds together you just have that number of surds. E.g.

√2 + √2 + √2= 3√2

If a surd has a square number as a factor you can use law 1 and/or law 2 and work backwards to take that out and simplify the surd. E.g. √500 = √100 x √5 = 10√5

Rationalization of surds

Surds may also appear in fractions. Rationalizing the denominator of such a fraction means finding an equivalent fraction that does NOT have a surd on the bottom of the fraction (though it CAN have a surd on the top!).

If the surd contains a square root by itself or a multiple of a square root, to get rid of it, you must multiply BOTH the top and bottom of the fraction by that square root value.

e.g. 6 x √7 = 6√7

√7 x √7 7

e.g.2 6 + √2 x √3 = 6√3 + √2 x √3 = 6√3 + √6

2√3 x √3 2 x √3 x √3 6

i.e. 2 x 3

If the surd on the bottom involves addition or subtraction with a square root, to get rid of the square root part you must use the ‘difference of two squares’ and multiply BOTH the top and bottom of the fraction by the bottom surd’s expression but with the inverse operation.

e.g.3 7 x (2 – √2) = 14 – 7√2 = 14 – 7√2

2 + √2 x (2 – √2) 2² – (√2)² 2

i.e. 4 – 2

Notes on the ‘Difference of two squares’…

Squaring… (2 + √2)(2 + √2) = 2(2 + √2) + √2(2 + √2)

(ops the same) = 4 + 2√2 + 2√2 + √2√2

= 4 + 4√2 + 2 = 6 + √2 (still a surd)

Multiplying… (2 + √2)(2 – √2) = 2(2 – √2) + √2(2 – √2)

(opposite ops) = 4 – 2√2 + 2√2 – √2√2

= 4 (cancel out) – 2 = 2 (not a surd)

In essence, as long as the operation in each brackets is the opposite, the middle terms will always cancel each other out and you will be left with the first term squared subtracting the second term squared.

i.e. (5 + √7)(5 – √7) à 5² – (√7)² = 25 – 7 = 18

Example

Simplify by rationalizing the denominator

Solution

Note

If the product of the two surds gives a rational number then the product of the two surds gives conjugate surds.

End of topic

Past KCSE Questions on the topic.

1. Without using logarithm tables, find the value of x in the equation

Log x³ + log 5x = 5 log2 – log 2 5

2. Simplify (1 ÷ √3) (1 – √3)

Hence evaluate 1 to 3 s.f. given that √3 = 1.7321

1 + √3

3. If √14 – √ 14 = a√7 + b√2

√7-√2 √ 7 + √ 2

 Find the values of a and b where a and b are rational numbers.

4. Find the value of x in the following equation 49^(x+1) + 7^(2x) = 350

5. Find x if 3 log 5 + log x² = log 1/125

6. Simplify as far as possible leaving your answer inform of a surd

 1 – 1

 √14 – 2 √3 √14 + 2 √3

7. Given that tan 75⁰ = 2 + √3, find without using tables tan 15⁰ in the form p+q√m, where p, q and m are integers.

8. Without using mathematical tables, simplify

63 + 72

 32 + 28

9. Simplify 3 + 1 leaving the answer in the form a + b Öc, where a, b and c Ö5 -2 Ö5 are rational numbers