Powers and Roots Class 6 ICSE — Chapter 7
The powers and roots Class 6 ICSE chapter, Chapter 7 of the ICSE Mathematics book, teaches you how to write repeated multiplication in short form and how to undo it. You will learn exponential notation, three exponent laws, squares, cubes, square roots, and cube roots. Three methods appear for each root: prime factors, fraction form, and index form. The chapter closes each method with a set of exercises.
Download Powers and Roots Class 6 ICSE Mathematics Chapter 7 PDF
Download the Powers and Roots Class 6 ICSE Mathematics chapter PDF to revise offline or print for practice. It is the full chapter as printed in the book — 13 pages, 6 sections, 48 questions.
What is inside Chapter 7
13 pages · 6 sections · 14 worked examples · 48 exercise questions
| Section | Title |
|---|---|
| 7.1 | EXPONENTIAL NOTATIONS (Positive powers only) |
| 7.2 | PROPERTIES OF EXPONENTS |
| 7.7 | CUBES |
| 7.8 | CUBE ROOT |
| 7.9 | SYMBOL FOR THE CUBE ROOT |
| 7.10 | METHODS OF FINDING THE CUBE ROOT |
- EXERCISE 7(A): 8 questions
- EXERCISE 7(B): 5 questions
- EXERCISE 7(C): 11 questions
- EXERCISE 7(D): 7 questions
- EXERCISE 7(E): 9 questions
- Revision Exercise (Chapter 7): 8 questions
7.1 EXPONENTIAL NOTATIONS (Positive powers only)
Exponential notation is the compact way to show a number multiplied by itself several times. Instead of writing 3 \times 3 \times 3 \times 3, you write 3^4. The repeating number 3 is the base, and 4 is the exponent, also called the index or power. You read 3^4 as “3 raised to the power 4” or “the fourth power of three.”
Two rules hold for powers. For any non-zero number a, a^0 = 1. So 2^0 = 1, (-5)^0 = 1, and \left(\frac{2}{3}\right)^0 = 1. For any number a, a^1 = a. So 5^1 = 5, (-2)^1 = -2, and \left(\frac{2}{3}\right)^1 = \frac{2}{3}.
The position of the minus sign changes the result. (-3)^4 = (-3)(-3)(-3)(-3) = 81, but -(3)^4 = -(3 \times 3 \times 3 \times 3) = -81. Four negatives multiply to a positive. A minus outside the bracket stays negative.
Worked pattern with fresh numbers: (-2)^4 = (-2)(-2)(-2)(-2) = 16, while -(2)^4 = -16. The sign of the base decides the sign of the power. A positive base gives a positive result; a negative base to an even power also gives a positive result.
You will also evaluate expressions with variables. For 6a^2b^3 with a = 3 and b = 2, substitute and multiply: 6 \times 3 \times 3 \times 2 \times 2 \times 2 = 432. A table in this section rewrites numbers like 72 as prime factor products, 72 = 2^3 \times 3^2. Exercise 7(A) has 8 questions on these skills.
7.2 PROPERTIES OF EXPONENTS
Three properties let you combine powers quickly. Product Law: a^m \times a^n = a^{m+n}. For example, 3^7 \times 3^3 = 3^{10}. Quotient Law: \frac{a^m}{a^n} = a^{m-n} when m \gt n, and \frac{a^m}{a^n} = \frac{1}{a^{n-m}} when m \lt n. So \frac{3^7}{3^4} = 3^3, while \frac{3^4}{3^7} = \frac{1}{3^3}. Power Law: (a^m)^n = a^{m \times n}. So (a^5)^3 = a^{15}.
The first two laws need the same base. The power law distributes over multiplication: (a \times b)^5 = a^5 \times b^5. It does not distribute over addition or subtraction: (a + b)^5 \neq a^5 + b^5 and (a - b)^5 \neq a^5 - b^5. This is a common mistake in this ICSE chapter.
Then the chapter turns to squares. A square is a number multiplied by itself. Since 4 \times 4 = 16, we write (4)^2 = 16. The square of any number, positive or negative, is always positive: (-3)^2 = 9. The square root reverses the square: the square root of 16 is 4, because 4 \times 4 = 16. The radical sign \sqrt{} denotes it, so \sqrt{9} = 3.
Method 1 — prime factors: express the number as a product of primes, group identical factors in twos, take one from each pair, and multiply. For 324: 324 = 2 \times 2 \times 3 \times 3 \times 3 \times 3, so \sqrt{324} = 2 \times 3 \times 3 = 18.
Method 2 — fraction form: take the square root of numerator and denominator separately. \sqrt{\frac{36}{49}} = \frac{\sqrt{36}}{\sqrt{49}} = \frac{6}{7}.
Method 3 — index form: keep the base, divide the index by 2. This works only when the index is even. \sqrt{2^{10}} = 2^{10 \div 2} = 2^5 = 32.
Exercise 7(B) has 5 questions on the exponent laws, Exercise 7(C) has 11 on squares, and Exercise 7(D) has 7 on square roots. The earlier chapters in the Class 6 Maths book list build the multiplication skills these methods need.
7.7 CUBES
A cube is a number multiplied by itself three times. Since 4 \times 4 \times 4 = 64, we write (4)^3 = 64. The cube of a positive number is always positive: (3)^3 = 27. The cube of a negative number is always negative: (-3)^3 = -27. Three negatives multiply to a negative.
The chapter shows that -a^3 = (-a)^3. So -4^3 = (-4)^3 = -64, and -6^3 = (-6)^3 = -216. Both forms give the same value because the exponent applies to the base before the minus sign acts.
Worked patterns with fresh numbers: the cube of a fraction \left(\frac{3}{4}\right)^3 = \frac{3}{4} \times \frac{3}{4} \times \frac{3}{4} = \frac{27}{64}. The cube of a decimal (0.4)^3 = 0.4 \times 0.4 \times 0.4 = 0.064. The cube of a mixed number \left(1\frac{1}{2}\right)^3 = \left(\frac{3}{2}\right)^3 = \frac{27}{8}.
Exercise 7(E) has 9 questions on cubing whole numbers, fractions, decimals, and negative numbers.
7.8 CUBE ROOT
The cube root of a given number is that number which, multiplied by itself three times, gives the given number. Since 4 \times 4 \times 4 = 64, the cube root of 64 is 4. Since \frac{2}{3} \times \frac{2}{3} \times \frac{2}{3} = \frac{8}{27}, the cube root of \frac{8}{27} is \frac{2}{3}.
The sign carries through in this Class 6 ICSE chapter. The cube root of a negative number is negative: since (-2)^3 = -8, the cube root of -8 is -2. Since (-0.3)^3 = -0.027, the cube root of -0.027 is -0.3. A positive number has a positive cube root; a negative number has a negative cube root.
This definitional section sets up the working methods in section 7.10. Before computing a cube root, check the sign of the given number so you know whether the answer is positive or negative.
7.9 SYMBOL FOR THE CUBE ROOT
The cube root uses the radical sign with a small 3 inside: \sqrt[3]{\phantom{x}}. The cube root of 243 is written \sqrt[3]{243} or in fractional index form (243)^{\frac{1}{3}}. Both notations mean the same thing and give the same value.
For example, \sqrt[3]{\frac{27}{64}} = \frac{3}{4} and \left(\frac{27}{64}\right)^{\frac{1}{3}} = \frac{3}{4}. A negative radicand keeps a negative cube root: \sqrt[3]{-8} = -2.
You may see either notation in the exercises. The radical form \sqrt[3]{} appears most often, while the index form ()^{\frac{1}{3}} links back to the exponential notation from section 7.1. Section 7.10 shows the methods for computing these values.
7.10 METHODS OF FINDING THE CUBE ROOT
Method 1 — factor method (four steps): express the number as a product of prime factors; group identical factors in threes; take one factor from each group; multiply those factors. For 512: 512 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2, so \sqrt[3]{512} = 2 \times 2 \times 2 = 8. For a negative radicand, \sqrt[3]{-343} = -7, because the cube root of a negative number is negative.
Method 2 — fraction form: take the cube root of numerator and denominator separately. \sqrt[3]{\frac{64}{125}} = \frac{\sqrt[3]{64}}{\sqrt[3]{125}} = \frac{4}{5}.
Method 3 — index form: keep the base, divide the index by 3. This works only when the index is exactly divisible by 3. \sqrt[3]{2^{9}} = 2^{9 \div 3} = 2^3 = 8. For a product, divide each index: \sqrt[3]{3^6 \times 2^3} = 3^2 \times 2^1 = 18.
The cube root distributes over multiplication but not over addition or subtraction. \sqrt[3]{a^3 b^3} = ab, but \sqrt[3]{a^3 + b^3} \neq a + b. The Revision Exercise for Chapter 7 has 8 questions covering the whole chapter, and the fraction-form roots reuse methods from the decimal fractions chapter.
What to practise from Powers and Roots Class 6 ICSE
Each exercise block drills one skill from this chapter. Work through them in order, matching each to the method it practises.
- Exercise 7(A) — 8 questions on exponential notation, bases, exponents, and simplifying products.
- Exercise 7(B) — 5 questions on the product, quotient, and power laws.
- Exercise 7(C) — 11 questions on finding squares of whole numbers, fractions, decimals, and negative numbers.
- Exercise 7(D) — 7 questions on square roots by prime factors, fraction form, and index form.
- Exercise 7(E) — 9 questions on cubes of whole numbers, fractions, decimals, and negative numbers.
- Revision Exercise (Chapter 7) — 8 questions covering powers, roots, and mixed evaluations.
All exercises follow the CISCE-aligned Class 6 Maths curriculum. The official CISCE website lists the current syllabus details.
Powers and Roots Class 6 ICSE frequently asked questions
What is the difference between (-3)^4 and -(3)^4 in powers and roots class 6 icse?
(-3)^4 = 81 because four negative threes multiply to a positive. -(3)^4 = -81 because the minus sits outside the bracket and stays negative after the power is computed.
Why does any number raised to the power 0 equal 1 in this Class 6 ICSE chapter?
For any non-zero number a, a^0 = 1. This follows from the quotient law: \frac{a^m}{a^m} = a^{m-m} = a^0, and any number divided by itself equals 1. So 2^0 = 1, (-5)^0 = 1, and \left(\frac{2}{3}\right)^0 = 1.
How do I find the square root of a fraction in Class 6 ICSE Maths?
Take the square root of the numerator and the square root of the denominator separately. \sqrt{\frac{36}{49}} = \frac{\sqrt{36}}{\sqrt{49}} = \frac{6}{7}.
How do I find the cube root of a negative number in powers and roots?
The cube root of a negative number is negative. Since (-7)^3 = -343, the cube root of -343 is -7. Factor the positive part, group in threes, take one from each group, and keep the negative sign.
Why can’t I split (a + b)^2 into a^2 + b^2 in this ICSE chapter?
Powers distribute over multiplication only, never over addition or subtraction. (a + b)^2 = (a+b)(a+b) = a^2 + 2ab + b^2, not a^2 + b^2. The same restriction holds for cube roots: \sqrt[3]{a^3 + b^3} \neq a + b.
Reference: ICSE Class 6 Mathematics textbook material, chapter 7: Powers and Roots — CISCE-aligned.