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About Indices (Exponents) — Class 9 ICSE

Apply laws of indices to simplify expressions with fractional and negative exponents. This topic is part of the ICSE Class 9 mathematics syllabus (chapter: Chapter 6). On this page you can practice 50 questions across three difficulty levels — 20 easy, 19 medium, and 11 hard — each with a visual step-by-step solution, plus a timed 28-question mastery test with a live leaderboard. Worked examples and sample questions from the topic are below.

What you'll learn in Indices (Exponents)

  • Introduction to Indices (Exponents)
  • Fundamental Laws of Indices
  • Zero and Negative Integer Exponents
  • Fractional Exponents (Roots)
  • Combining Laws and Advanced Simplification

Interactive lesson · about 15 minutes · checkpoint question after every unit

Indices (Exponents) — solved examples for Class 9 ICSE

Example 1easy

Rohan simplified (5^0 × 2^-1) + (3^2 × 1^0) as follows:
Step 1: (1 × 1/2) + (9 × 0) = 1/2 + 0.
Step 2: Result = 1/2.
Where did Rohan make a mistake?
  1. A)He incorrectly evaluated 5^0.
  2. B)He incorrectly evaluated 2^-1.
  3. C)He incorrectly evaluated 1^0.
  4. D)He correctly solved the problem, there is no mistake.

Step-by-step solution

  1. The rule for zero exponent states that any non-zero number raised to the power of zero is 1 (a^0 = 1).
  2. Therefore, 1^0 should be 1, not 0.
  3. Rohan made a mistake in evaluating 1^0 as 0. The correct calculation would be (1 × 1/2) + (9 × 1) = 1/2 + 9 = 19/2.

Answer: He incorrectly evaluated 1^0.

Example 2medium

Simplify the expression: [ (x^(a-b))^3 × (x^(b-c))^3 × (x^(c-a))^3 ]
  1. A)x^0
  2. B)x^1
  3. C)x^(a+b+c)
  4. D)x^3

Step-by-step solution

  1. Apply the law (a^m)^n = a^(m×n) to each term:
    (x(a−b))3=x(3a−3b);(x(b−c))3=x(3b−3c);(x(c−a))3=x(3c−3a)(x^(a-b))^3 = x^(3a-3b); (x^(b-c))^3 = x^(3b-3c); (x^(c-a))^3 = x^(3c-3a)
  2. Now, multiply the terms. Using a^m × a^n = a^(m+n), add the exponents:
    x(3a−3b)×x(3b−3c)×x(3c−3a)=x((3a−3b)+(3b−3c)+(3c−3a))x^(3a-3b) × x^(3b-3c) × x^(3c-3a) = x^((3a-3b) + (3b-3c) + (3c-3a))
  3. Combine the exponents:
    x(3a−3b+3b−3c+3c−3a)=x0x^(3a - 3b + 3b - 3c + 3c - 3a) = x^0
  4. Any non-zero number raised to the power of 0 is 1.
    x0=1x^0 = 1

Answer: x^0

Example 3hard

Simplify the expression: (16/81)^(-3/4) × (25/9)^(3/2) ÷ (5/2)^3
  1. A)1
  2. B)2/3
  3. C)4/9
  4. D)8/27

Step-by-step solution

  1. First term: (16/81)^(-3/4) = (81/16)^(3/4) = ((3/2)^4)^(3/4) = (3/2)^3 = 27/8.
  2. Second term: (25/9)^(3/2) = ((5/3)^2)^(3/2) = (5/3)^3 = 125/27.
  3. Third term: (5/2)^3 = 125/8.
  4. Combine the simplified terms: (27/8) × (125/27) ÷ (125/8) = (27/8) × (125/27) × (8/125).
  5. Cancel common factors: (27/27) × (125/125) × (8/8) = 1 × 1 × 1 = 1.

Answer: 1

Practice questions on Indices (Exponents)

  1. Q1.easy

    Which of the following statements about exponents is true?
    1. A)(-3)^-2 = -1/9
    2. B)2^-3 = -8
    3. C)(1/2)^-2 = 4
    4. D)-5^-1 = 1/5
    Show answer

    Answer: (1/2)^-2 = 4

    Hint: Recall how negative exponents work, especially with fractions and negative bases.

  2. Q2.easy

    Simplify: ( (2^2)^-1 / 2^3 )^-2
    1. A)2^10
    2. B)2^-10
    3. C)2^5
    4. D)2^-5
    Show answer

    Answer: 2^10

    Hint: Apply the power rule ((a^m)^n = a^(mn)) first, then the quotient rule (a^m / a^n = a^(m-n)).

  3. Q3.easy

    Evaluate: (16)^(3/4) × (81)^(-1/4)
    1. A)2/3
    2. B)8/3
    3. C)16/3
    4. D)4/3
    Show answer

    Answer: 8/3

    Hint: Express the base numbers as powers of prime numbers before applying the fractional exponents.

  4. Q4.medium

    If (2^(n-1) × 4^(2n+1)) / (8^(n+1)) = 1/4, find the value of n.
    1. A)-2
    2. B)-1
    3. C)0
    4. D)1
    Show answer

    Answer: -1

    Hint: Express all bases as powers of 2 first. Then apply the laws of exponents to simplify the expression and equate the powers.

  5. Q5.medium

    Ravi was asked to simplify (64)^(-1/3) ÷ (27)^(-1/3). He wrote the steps as follows:
    Step 1: (64)^(-1/3) ÷ (27)^(-1/3) = (1/64)^(1/3) ÷ (1/27)^(1/3)
    Step 2: = (1/4) ÷ (1/3)
    Step 3: = 1/4 × 3/1
    Step 4: = 3/4
    Is there any mistake in Ravi's solution?
    1. A)Yes, in Step 1
    2. B)Yes, in Step 2
    3. C)No, the solution is correct
    4. D)Yes, in Step 3
    Show answer

    Answer: No, the solution is correct

    Hint: Carefully review each step for the correct application of negative and fractional exponents. Remember that a^(-m) = 1/a^m and a^(1/n) is the nth root of a.

  6. Q6.medium

    Which of the following statements is TRUE?
    1. A)(a^m)^n = a^(m+n)
    2. B)a^m × a^n = a^(m×n)
    3. C)a^m + a^n = a^(m+n)
    4. D)(a^m)^n = a^(m×n)
    Show answer

    Answer: (a^m)^n = a^(m×n)

    Hint: Recall the fundamental laws of exponents. Each option represents a common mistake or a correct application of a law. Focus on the power of a power rule.

  7. Q7.hard

    If 5^(x-3) × 25^(x-1) = 1 / (125^(2x-3)), what is the value of x?
    1. A)7/9
    2. B)14/9
    3. C)11/9
    4. D)13/9
    Show answer

    Answer: 14/9

    Hint: Express all terms with the same base (5) and then equate the exponents to form a linear equation.

  8. Q8.hard

    The expression (x^a / x^b)^(a+b) × (x^b / x^c)^(b+c) × (x^c / x^a)^(c+a) simplifies to:
    1. A)x^(a^2+b^2+c^2)
    2. B)x^(abc)
    3. C)1
    4. D)x^(a+b+c)
    Show answer

    Answer: 1

    Hint: Apply the law (x^m / x^n) = x^(m-n) first, then use the difference of squares identity (u-v)(u+v) = u^2 - v^2.

  9. Q9.hard

    If (√(x^3 y^(-2)))^z simplifies to x^6 y^(-4), what is the value of z?
    1. A)2
    2. B)3
    3. C)4
    4. D)6
    Show answer

    Answer: 4

    Hint: Rewrite the square root as a fractional exponent (power of 1/2) and then apply the power of a power rule. Finally, equate the exponents of x (or y) on both sides.

These are 9 of the 50 questions available for Indices (Exponents). Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.