Number Systems opens Class 9 by widening your child’s world from rationals to real numbers — bringing in irrationals like √2, surds, the laws of exponents with fractional powers, and rationalising a denominator. The ideas are precise and reward neat working: knowing that √27 = 3√3, that 0.3̄ = 1/3, or that a rational times an irrational stays irrational turns fiddly-looking questions into quick ones. Short, regular practice is what makes surds feel friendly.

This free printable worksheet has 12 questions from Bmathpro’s own question bank of 10,500+ validated maths questions — the same bank our enrolled students practise from. Every answer comes with a one-line explanation, so you can check your child’s method, not just the final answer.

Set aside 15–20 minutes, keep the answers hidden until the end, and let your child attempt every question — a wrong attempt tells you more than a skipped one.

What this worksheet covers

• Decimal expansions, and rational vs irrational numbers
• Laws of exponents with fractional powers (like 64^(1/3))
• Simplifying surds (like √27 = 3√3)
• Converting a recurring decimal to a fraction
• Rationalising a denominator
• Finding a rational number between two rationals

Maps to the CBSE Class 9 Number Systems chapter (NCERT) and equivalent ICSE/State-board units.

Class 9 Number Systems Worksheet — 12 Questions

Warm-up

1. Write the decimal expansion of 1/4. (Write your answer.)

2. Find the value of (√3)². (Write your answer.)

3. Which of the following numbers is irrational?
A. 0.25  B. √9  C. √2  D. 3/7

Building up

4. Evaluate using laws of exponents: 64^(1/3). (Write your answer.)

5. Simplify √27 into the form a√3. What is a? (Write your answer.)

6. Express the repeating decimal 0.333… (i.e. 0.3̄) as a fraction in simplest form. (Write your answer.)

7. Evaluate using laws of exponents: 125^(2/3). (Write your answer.)

8. The decimal expansion of a rational number is always:
A. always terminating  B. terminating or non-terminating recurring  C. always non-terminating  D. non-terminating and non-recurring

Challenge

9. Find a rational number lying exactly halfway between 1/4 and 1/2. (Write your answer.)

10. Simplify: √45 + √20. Give the answer in simplest surd form. (Write your answer.)

11. Rationalise the denominator and simplify: 1 / (√5 − 2). Give the answer in the form √5 + a. (Write your answer.)

12. The product of a non-zero rational number and an irrational number is always:
A. rational  B. always an integer  C. irrational  D. sometimes rational, sometimes irrational

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Answer key with explanations

Show answers and explanations (parents — open this after your child finishes)

1. 0.25 — 1 ÷ 4 = 0.25.
2. 3 — squaring undoes the square root: (√3)² = 3.
3. C. √2 — cannot be written as a ratio of integers; the others are rational.
4. 4 — 64^(1/3) = (4³)^(1/3) = 4.
5. 3 — √27 = √(9×3) = 3√3, so a = 3.
6. 1/3 — 10x − x = 3 → 9x = 3 → x = 1/3.
7. 25 — 125^(2/3) = (5³)^(2/3) = 5² = 25.
8. B. terminating or non-terminating recurring — every rational’s decimal stops or repeats.
9. 3/8 — average = (3/4) ÷ 2 = 3/8.
10. 5√5 — √45 = 3√5, √20 = 2√5, sum = 5√5.
11. √5 + 2 — multiply by (√5 + 2)/(√5 + 2); denominator = 5 − 4 = 1.
12. C. irrational — a non-zero rational times an irrational stays irrational.

How to use this worksheet at home

• Keep it short and calm. 15–20 minutes beats an hour of pressure.
• Simplify the surd first. On √-questions, ask your child to pull out the largest square factor before doing anything else — √27 becomes 3√3 and the rest falls into place.
• Praise the method, not the marks. (We deliberately don’t put marks on our worksheets.)

FAQs

Is this worksheet free to print?
Yes — free to use and print at home, no sign-up needed.

Which syllabus does it follow?
Standard Class 9 Number Systems chapter — CBSE (NCERT), ICSE and State boards.

How do you rationalise a denominator with a surd?
Multiply the top and bottom by the conjugate (for √5 − 2, that is √5 + 2). The denominator becomes a difference of squares with no surd left. Question 11 practises exactly this — a free demo class shows the full working.

Where can my child get more practice like this?
Enrolled students get Math Gym, our paid practice app with 10,500+ questions — see our courses page.

Also practise: Polynomials · Number Systems · Linear Equations in Two Variables · all Class 9 worksheets.

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