Nalanda College, Colombo 10 Grade 11 Maths Worksheet: 1 Real Numbers

Add like surds with the same root

Answer: 6√3

💡 What it's asking: Surds with the same number under the root work like like terms in algebra. Just add the numbers in front and keep √3.

1. 1Notice both terms have √3, so they are like terms (same kind).
2. 2Think of √3 as a letter x: 4x + 2x = 6x.
3. 3Add the numbers in front: 4 + 2 = 6.
4. 4Keep the √3 as it is: the answer is 6√3.

✅ √3 is about 1.73; 4√3 + 2√3 ≈ 6.93 + 3.46 = 10.39, and 6 × 1.73 ≈ 10.39 too.

Collect two kinds of like surds

Answer: 8√3 + √2

💡 What it's asking: There are two kinds of surds here, √3 and √2. Collect each kind separately, like collecting x's and y's.

1. 1Pick out the √3 terms: 5√3 and 3√3.
2. 2Add them: 5 + 3 = 8, so 8√3.
3. 3Pick out the √2 terms: 2√2 and −√2 (which means −1√2).
4. 4Combine them: 2 − 1 = 1, so 1√2, written simply as √2.
5. 5Write the answer: 8√3 + √2. These are unlike surds, so stop here.

✅ Count: you started with 5 + 3 = 8 lots of √3 and 2 − 1 = 1 lot of √2, matching the answer.

Collect like surds in a four-term expression

Write an entire surd in simplest form

Change a surd into an entire surd

Simplify surds to a common root then combine

Multiply two surds with coefficients

Rationalise then substitute a given root value

Divide surds and simplify

Rationalise a surd denominator

Value of fractions with surd denominators using given roots

Simplify sums of surds after reducing each surd

Write entire surds in simplest surd form

Simplify expressions mixing surd fractions and surds

Multiply and divide surds

Pick the rational numbers from a mixed list

Write entire surds in simplest form

Simplify longer sums of surds

Rationalise denominators of two surd fractions