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Tamil Nadu (TNBSE)Class 8 Mathematics★ 8-12 marks in the TN Class 8 annual exam through MCQs and number, root and exponent problems

Numbers

Samacheer Kalvi (TN State Board) Class 8 Maths Chapter 1, Numbers. Rational numbers and their properties, square roots, cubes and cube roots, and exponents and powers (laws of exponents). Notes with formulas, worked examples, common mistakes, book-back practice and quick revision.

⏱ 22 min readMedium difficulty✓ Free · NCERT-aligned

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By the end of this chapter you'll be able to…

1Operate on rational numbers and use their properties
2Find square roots by factorisation and long division
3Find cubes and cube roots
4Apply the laws of exponents
5Write numbers in scientific notation

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Why this chapter matters

Numbers builds fluency with rational numbers, roots and powers — the backbone of all later mathematics. Square root, cube root and exponent problems are reliable scoring questions in the TN Class 8 exam.

Before you start — revise these

A 5-minute refresher here will save you 30 minutes of confusion below.

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Class 7 — Integers and fractions

Operations on integers and fractions.

Numbers — Class 8 Maths (Samacheer Kalvi)

TN State Board (Samacheer Kalvi) Class 8 Mathematics, Chapter 1. Rational numbers, roots and powers.

1. About this chapter

This chapter covers rational numbers and their properties, square roots, cubes and cube roots, and exponents and powers.

2. Rational numbers

A rational number is written as p/q (q ≠ 0). They can be added, subtracted, multiplied and divided.
Properties: closure, commutative, associative and distributive laws; 0 is the additive identity, 1 the multiplicative identity.
Between any two rational numbers there are infinitely many rationals.

3. Square roots, cubes and cube roots

Square root (√): for a perfect square, found by prime factorisation or long division.
Cube: n³; cube root (∛): found by grouping prime factors in threes.
A perfect cube has prime factors in groups of three (e.g., 27 = 3³, ∛27 = 3).

4. Exponents and powers (laws)

aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ.
a⁰ = 1, a⁻ⁿ = 1/aⁿ, (ab)ᵐ = aᵐbᵐ.
Scientific notation: m × 10ⁿ with 1 ≤ m < 10.

5. Worked examples

Example 1. Find √324 by prime factorisation. 324 = 2² × 3⁴ → √324 = 2 × 3² = 18.

Example 2. Find ∛216. 216 = 2³ × 3³ → ∛216 = 2 × 3 = 6.

Example 3. Simplify 2³ × 2⁴. = 2³⁺⁴ = 2⁷ = 128.

6. Common mistakes

Mistake: Writing a⁻ⁿ = −aⁿ. Fix: a⁻ⁿ = 1/aⁿ (a negative exponent means reciprocal).
Mistake: Adding exponents when bases differ. Fix: aᵐ × aⁿ = aᵐ⁺ⁿ only when the base is the same.
Mistake: Forgetting a⁰ = 1. Fix: Any non-zero number to the power 0 is 1.

7. Practice (book-back style)

Find √196.
Find ∛125.
Simplify 5⁴ ÷ 5².
Write 0.00056 in scientific notation.
Evaluate (2³)².

8. Answer key

196 = 2² × 7² → √196 = 2 × 7 = 14.
125 = 5³ → ∛125 = 5.
5⁴⁻² = 5² = 25.
5.6 × 10⁻⁴.
2⁶ = 64.

9. Quick revision

Chapter 1 · rational numbers, roots, powers.
Rational = p/q; properties: closure, commutative, associative, distributive.
√ by prime factorisation/long division; ∛ by grouping in threes.
Laws: aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ ÷ aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ; a⁻ⁿ = 1/aⁿ; a⁰ = 1.
Scientific notation: m × 10ⁿ, 1 ≤ m < 10.

Key formulas & results

Everything you need to memorise, in one card. Screenshot this for revision.

Laws of exponents

aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ ÷ aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ

Same base required for product/quotient.

Negative / zero exponent

a⁻ⁿ = 1/aⁿ; a⁰ = 1

For non-zero a.

Roots by factorisation

√ pairs of primes; ∛ triples of primes

Group prime factors.

Scientific notation

m × 10ⁿ, 1 ≤ m < 10

n is an integer.

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Common mistakes & fixes

These are the exact errors that cost students marks in board exams. Read them once, save yourself the trouble.

WATCH OUT

✗ Writing a⁻ⁿ = −aⁿ

✓ a⁻ⁿ = 1/aⁿ — a negative exponent means reciprocal.

WATCH OUT

✗ Adding exponents when bases differ

✓ aᵐ × aⁿ = aᵐ⁺ⁿ only when the base is the same.

WATCH OUT

✗ Forgetting a⁰ = 1

✓ Any non-zero number to the power 0 is 1.

NCERT exercises (with solutions)

Every NCERT exercise from this chapter — what it covers and how many questions to expect.

Multiple choice questions

MCQs on rational numbers, roots and exponents.

Questions

Root problems

Square roots and cube roots by factorisation.

Questions

Exponent problems

Laws of exponents and scientific notation.

Questions

Practice problems

Try each one yourself before tapping "Show solution". Active recall > rereading.

Q1EASY· Numerical

Find √196.

▶ Show solution

196 = 2² × 7² → √196 = 14.

Q2EASY· Numerical

Find ∛125.

▶ Show solution

125 = 5³ → ∛125 = 5.

Q3EASY· Exponents

Simplify 5⁴ ÷ 5².

▶ Show solution

5⁴⁻² = 5² = 25.

Q4EASY· Notation

Write 0.00056 in scientific notation.

▶ Show solution

5.6 × 10⁻⁴.

Q5MEDIUM· Numerical

Find √324 by prime factorisation.

▶ Show solution

324 = 2² × 3⁴ → √324 = 2 × 3² = 18.

Q6EASY· Exponents

Evaluate (2³)².

▶ Show solution

2⁶ = 64.

5-minute revision

The whole chapter, distilled. Read this the night before the exam.

•Chapter 1 of Samacheer Kalvi Class 8 Mathematics.
•Rational = p/q; closure, commutative, associative, distributive.
•√ by pairing primes; ∛ by grouping in threes.
•aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ ÷ aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ.
•a⁻ⁿ = 1/aⁿ; a⁰ = 1.
•Scientific notation: m × 10ⁿ, 1 ≤ m < 10.

Tamil Nadu (TNBSE) marks blueprint

Where the marks come from in this chapter — so you can plan your prep.

Typical chapter weightage: 8-12 marks across MCQ, root and exponent problems

Question type	Marks each	Typical count	What it tests
MCQ	1	1-3	Rational numbers, roots, exponents
Roots	2-3	1-2	Square and cube roots
Exponents	2-3	1-2	Laws and scientific notation

Prep strategy

Practise prime factorisation for roots
Memorise the laws of exponents
Convert to and from scientific notation
Use rational-number properties

Where this shows up in the real world

This chapter isn't just an exam topic — it lives in the world around you.

Measurement

Square and cube roots relate areas and volumes to lengths.

Science notation

Exponents express very large and small quantities.

Finance

Rational numbers handle money calculations.

Exam strategy

Battle-tested tips from teachers and toppers for this chapter.

Show prime factorisation for roots
Apply exponent laws step by step
Keep 1 ≤ m < 10 in scientific notation
Use reciprocal for negative exponents

Going beyond the textbook

For olympiad aspirants and curious learners — topics that build on this chapter.

Find the smallest number to multiply 1080 to make it a perfect cube.
Simplify an expression with mixed positive and negative exponents.

Where else this chapter is tested

CBSE board isn't the only one — other exams test this chapter too.

TN Class 8 Annual ExamHigh

Foundation / NMMS MathematicsMedium

School unit testsHigh

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Questions students ask

The real ones — pulled from the Q&A community and tutor sessions.

Write the number as a product of primes, group the identical primes in threes, and take one factor from each group; their product is the cube root.

It means the reciprocal: a⁻ⁿ = 1/aⁿ, so a smaller (or negative) power gives a fraction.

Practice more, ask doubts, find a tutor

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