Indices Class 9 ICSE — ICSE Maths Chapter 7
Indices class 9 ICSE is the chapter that teaches you how to work with powers and exponents. In the Class 9 ICSE Mathematics book, Chapter 7 defines what an index is, sets out the three laws of indices, and then shows how to handle fractional, negative and zero powers. This page walks through each section in order, with the formulas and worked patterns you can follow while the chapter is open in front of you.
Download Indices Exponents Class 9 ICSE Mathematics Chapter 7 PDF
Download the Indices Exponents Class 9 ICSE Mathematics chapter PDF to revise offline or print for practice. It is the full chapter as printed in the book — 7 pages, 5 sections, 26 questions.
What is inside Chapter 7
7 pages · 5 sections · 6 worked examples · 26 exercise questions
| Section | Title |
|---|---|
| 7.1 | INTRODUCTION |
| 7.2 | LAWS OF INDICES |
| 7.3 | HANDLING POSITIVE, FRACTIONAL, NEGATIVE AND ZERO INDICES |
| 7.4 | SIMPLIFICATION OF EXPRESSIONS |
| 7.5 | USING LAWS OF EXPONENTS |
- EXERCISE 7 (B): 14 questions
- EXERCISE 7 (C): 12 questions
7.1 INTRODUCTION
An index is the small number that tells you how many times to multiply a base by itself. If m is a positive integer, then a \times a \times a \times \ldots up to m terms is written as a^m. Here a is called the base, and m is called the power, the exponent, or the index.
The symbol a^m is read as “a power m” or “a raised to the power m“. The plural of index is indices, which gives the chapter its name. This compact notation turns long repeated multiplication into a single symbol, which makes later algebra far easier to read.
Take 5 \times 5 \times 5 \times 5. That is 5 written four times, so it equals 5^4. The base is 5 and the power is 4. You would read it as “5 raised to the power 4”.
7.2 LAWS OF INDICES
Once you know what an index means, the chapter states the three core laws that govern how indices combine. These are the rules you will apply in every later section and in both exercise sets. Each law needs the same base on both sides.
| Law | Rule | When it applies |
|---|---|---|
| 1st Law (Product Law) | a^m \times a^n = a^{m+n} | Multiplying two powers of the same base — add the exponents |
| 2nd Law (Quotient Law) | \frac{a^m}{a^n} = a^{m-n} | Dividing two powers of the same base — subtract the exponents |
| 3rd Law (Power Law) | (a^m)^n = a^{mn} | A power raised to another power — multiply the exponents |
The product law works because multiplying a m times and then n more times gives m+n copies in total. The quotient law subtracts because cancelling common factors of a leaves the difference of the exponents. The power law multiplies because (a^m)^n is the whole block a^m repeated n times.
All three laws hold for negative exponents as well as positive ones. The book applies the same rules to negative powers without any change.
Here is one pattern for each law, with fresh numbers. a^5 \times a^{-2} = a^{5-2} = a^3 uses the product law. \frac{b^8}{b^3} = b^{8-3} = b^5 uses the quotient law. (c^2)^4 = c^{2 \times 4} = c^8 uses the power law.
7.3 HANDLING POSITIVE, FRACTIONAL, NEGATIVE AND ZERO INDICES
This section extends the laws to every kind of index: positive, fractional, negative, and zero. Five rules cover the new cases, and each builds on the three laws from 7.2.
Rule 1 — a product or quotient under one power splits. (a \times b)^m = a^m \times b^m and \left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}. For example, (2 \times 3)^5 = 2^5 \times 3^5.
Rule 2 — a root becomes a fractional index. If a \neq 0 and n is a positive integer, then \sqrt[n]{a} = a^{1/n}. So \sqrt{a} = a^{1/2} and \sqrt[3]{a} = a^{1/3}.
Rule 3 — the general fractional form. a^{m/n} = \sqrt[n]{a^m}, and the converse \sqrt[n]{a^m} = a^{m/n} also holds. So 5^{2/3} = \sqrt[3]{5^2} and \sqrt[3]{a^5} = a^{5/3}.
Rule 4 — a negative index means the reciprocal. For any non-zero a, a^{-n} = \frac{1}{a^n} and a^n = \frac{1}{a^{-n}}. So a^{-3} = \frac{1}{a^3}.
Rule 5 — the zero index. Any non-zero number raised to the power zero equals one: a^0 = 1. The base must not be zero, because 0^0 is not covered by this rule.
A negative base changes sign with the parity of the power. (-a)^m = a^m when m is even, and (-a)^m = -a^m when m is odd. So (-2)^4 = 2^4 = 16 but (-2)^5 = -2^5 = -32.
Evaluate 8^{2/3} using Rule 3. Write it as (\sqrt[3]{8})^2. Since \sqrt[3]{8} = 2, the result is 2^2 = 4.
7.4 SIMPLIFICATION OF EXPRESSIONS
This section applies every law from 7.2 and 7.3 to evaluate and simplify full expressions. The standard method is to rewrite each base as a prime power first, then apply the power law, then combine the like bases. Board style questions in this ICSE chapter expect exactly that order of steps. You can browse the full Class 9 ICSE Maths book to see where this chapter sits next to others.
Prime factorisation comes first. A value like 432 factors as 2^4 \times 3^3. Once every base is a prime power, you handle the powers of 2 and 3 separately and then multiply the results.
Evaluate 16^{3/4}. Rewrite 16 as 2^4, so 16^{3/4} = (2^4)^{3/4} = 2^{4 \times 3/4} = 2^3 = 8.
Evaluate 25^{1/2} - 8^{-2/3}. The first term is (5^2)^{1/2} = 5. The second term is a negative fractional index: 8^{-2/3} = \frac{1}{8^{2/3}} = \frac{1}{(\sqrt[3]{8})^2} = \frac{1}{2^2} = \frac{1}{4}. The result is 5 - \frac{1}{4} = \frac{19}{4}.
A common mistake is mishandling a negative fractional index. Students often drop the minus sign and compute 8^{2/3} = 4, forgetting the reciprocal. The correct step is to take the reciprocal first: 8^{-2/3} = \frac{1}{8^{2/3}} = \frac{1}{4}.
The book also factors out a common power to simplify quotients. For 3^{a+2} - 3^{a+1}, write both terms with 3^a: 3^a \cdot 3^2 - 3^a \cdot 3^1 = 3^a(9 - 3) = 3^a \times 6. In a quotient, the 3^a factor cancels and leaves a plain number.
7.5 USING LAWS OF EXPONENTS
This final section uses the laws to solve exponential equations and to prove identities. The book shows three methods, each with a different purpose. The first two solve for x; the third proves a stated relation.
Method 1 — match the bases, then equate exponents. When both sides share a base, the exponents must be equal. Solve 4^x = 16^2. Rewrite both sides with base 2: (2^2)^x = (2^4)^2, so 2^{2x} = 2^8. The bases match, so 2x = 8 and x = 4.
Method 2 — substitute to form a quadratic. Some equations do not share a base at first. A form like k^{2x} + b \cdot k^x + c = 0 becomes a quadratic in y = k^x, because k^{2x} = (k^x)^2 = y^2. Solve the quadratic for y, then convert each value back to an equation in x.
The book demonstrates this on 2^{2x+3} - 9 \times 2^x + 1 = 0, where 2^{2x+3} = 8 \times 2^{2x} = 8y^2.
Method 3 — prove relations by chaining substitutions. Suppose p = q^{2a} and q = r^{2b}. Then p = (r^{2b})^{2a} = r^{4ab}. Adding r = p^{2c} gives p = (p^{2c})^{4ab} = p^{8abc}, so 8abc = 1. The same chain logic appears in both exercise sets. These exponent laws lead directly into Chapter 8 on logarithms, the inverse operation of an index.
Continue with the Class 9 ICSE Maths chapter on logarithms to see the connection.
What to practise from Indices Class 9 ICSE
The chapter closes with two exercise blocks. Work through them in order, since the second assumes the first. Both sets follow the syllabus published by the Council for the Indian School Certificate Examinations, whose official site is cisce.org.
- EXERCISE 7 (B) — 14 questions: solving for x, finding values of m and n, and proving identities.
- EXERCISE 7 (C) — 12 questions: evaluating expressions, simplifying quotients, and proving relations.
Both sets mix the three methods from 7.5. Expect equations where you match bases, expressions where you prime-factorise, and identities you must prove from scratch.
Indices Class 9 ICSE frequently asked questions
What are the three laws of indices in class 9 ICSE?
The three laws are the product law a^m \times a^n = a^{m+n}, the quotient law \frac{a^m}{a^n} = a^{m-n}, and the power law (a^m)^n = a^{mn}. Each applies when the base is the same on both sides. The book labels them the 1st, 2nd and 3rd laws in section 7.2.
How do I solve equations with indices class 9 ICSE when the bases are different?
Rewrite both sides with a common base first. For 4^x = 16^2, write 4 = 2^2 and 16 = 2^4, giving 2^{2x} = 2^8. Then equate exponents: 2x = 8, so x = 4. If no single base works, use substitution to form a quadratic.
What is the value of any non zero number raised to the power zero?
It is always 1. The rule a^0 = 1 holds for any non-zero base, so 5^0 = 1 and 2^0 = 1. The base must not be zero for this rule to apply.
How do negative and fractional indices work in class 9 ICSE?
A negative index means the reciprocal: a^{-n} = \frac{1}{a^n}. A fractional index combines a root and a power: a^{m/n} = \sqrt[n]{a^m}. For example, 8^{2/3} = (\sqrt[3]{8})^2 = 4.
Reference: ICSE Class 9 Mathematics textbook material, chapter 7: Indices Exponents — CISCE-aligned.