Surds

Syllabus tag: KCSE | Mathematics | Form 3 | Topic 4 Surds

Lesson objectives

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

• Define rational and irrational numbers.
• Simplify expressions with surds.
• Add, subtract, multiply and divide surds.
• Rationalise denominators containing surds.

Surds

A surd is a root that cannot be written exactly as a fraction, such as √2 or √5.

a) Rational and irrational

A rational number can be written as a fraction of two integers. All terminating and recurring decimals are rational.

An irrational number cannot. Its decimal never ends and never repeats.

So √4 is rational, since it equals 2. But √2 is irrational, and is a surd.

π is irrational too, but it is not a surd. It is not a root of a rational number.

b) Why leave answers in surd form

√2 written as a decimal is never exact, however many figures you give.

Leaving it as √2 keeps the answer exact. That matters in geometry and in later algebra.

c) The rules

√a × √b = √(ab).

√a ÷ √b = √(a/b).

But √(a + b) is not √a + √b. Test it. √(9 + 16) is √25 = 5, while √9 + √16 = 7.

That is the single most common error with surds.

d) Simplifying

Look for the largest square factor.

Split the root, take the square root of the square factor, and leave the rest inside.

For √72, the largest square factor is 36, giving 6√2. Using 4 instead gives 2√18, which is correct but not fully simplified.

e) Adding and subtracting

Only like surds can be combined, in the same way as like terms in algebra.

So 3√2 + 5√2 = 8√2. But 3√2 + 5√3 cannot be simplified.

Simplify each surd first. Terms that looked unlike may become like: √8 + √2 becomes 2√2 + √2 = 3√2.

f) Multiplying

Multiply the rational parts and the surd parts separately.

So 2√3 × 4√5 = 8√15.

Note that √a × √a = a. So 3√2 × 5√2 = 15 × 2 = 30, with no surd left.

g) Rationalising the denominator

It is conventional to remove surds from the denominator.

For a single surd, multiply top and bottom by that surd.

For a binomial denominator such as 2 + √3, multiply by its conjugate, 2 − √3.

The conjugate works because (a + b)(a − b) = a² − b², and squaring removes the surd.

So 1/(2 + √3) becomes (2 − √3)/(4 − 3), which is simply 2 − √3.

h) Where this is used

Exact answers in Pythagoras and trigonometry. Geometry involving diagonals of squares. Later work in calculus, where exact forms are required.

Words to know

• Rational number -- one expressible as a fraction of two integers.
• Irrational number -- one that cannot be so expressed; its decimal never terminates or recurs.
• Surd -- an irrational root left in root form.
• Like surds -- surds with the same number under the root sign.
• Conjugate -- the same two-term expression with the middle sign reversed.

:::checkpoint Check yourself

1. Simplify √50.
2. Simplify 2√12 + 3√3.
3. Rationalise 4/√7.
4. Why is √(9 + 16) not equal to √9 + √16? :::

Bridge to practice

The exercises begin with rational and irrational numbers and simplification, move through the four operations, and finish with rationalising both kinds of denominator. Simplify every surd fully before attempting to add or subtract — terms that look unlike often are not.