5. Exponents and Roots

CSS GSA — Paper 7, Part-II, Section VI, Sub-topic 5 (Exponents and Roots)

Past Paper Weightage

• Common in both direct short questions and as a step inside algebra/geometry problems — moderate-high frequency.

Key Laws of Exponents

$$a^m \times a^n = a^{m+n}$$ $$a^m \div a^n = a^{m-n}$$ $$(a^m)^n = a^{mn}$$ $$a^0 = 1 \quad (a \neq 0)$$ $$a^{-n} = \frac{1}{a^n}$$ $$(ab)^n = a^n b^n$$ $$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$$ $$a^{1/n} = \sqrt[n]{a}$$

Roots

• Square root (√): the number which, squared, gives the original — √49 = 7
• Cube root (∛): the number which, cubed, gives the original — ∛27 = 3
• Simplifying: √50 = √(25×2) = 5√2 (pull out perfect square factors)
• Rationalizing a denominator: multiply numerator and denominator by the conjugate/root to remove a root from the denominator, e.g., 1/√2 = √2/2

Worked Example

• 2³ × 2⁴ = 2⁷ = 128
• (3²)³ = 3⁶ = 729
• √72 = √(36×2) = 6√2

Past Paper Style Numericals (Solved)

1. Simplify: (2³ × 3²) ÷ (2 × 3)
   • Step 1: Compute the powers first: 2³ = 8, 3² = 9.
   • Step 2: Multiply: 8 × 9 = 72; and the denominator: 2 × 3 = 6.
   • Step 3: Divide: 72 ÷ 6 = 12.
2. If 3^x = 81, find x.
   • Step 1: Express 81 as a power of 3: 81 = 3×3×3×3 = 3⁴.
   • Step 2: Since 3^x = 3⁴, the exponents must be equal.
   • Step 3: Therefore x = 4.
3. Simplify: √45 + √20
   • Step 1: Break each root into a perfect-square factor times a remainder: √45 = √(9×5) = 3√5; √20 = √(4×5) = 2√5.
   • Step 2: Now both terms share the same radical (√5), so they can be combined like algebraic terms: 3√5 + 2√5.
   • Step 3: Add the coefficients: (3+2)√5 = 5√5.
4. Evaluate: (−64)^(1/3)
   • Step 1: A power of 1/3 means "cube root" — we need the number which, when cubed, gives −64.
   • Step 2: Test −4: (−4)³ = (−4)×(−4)×(−4) = 16×(−4) = −64. Matches.
   • Step 3: Answer: −4.
5. Simplify: (5²)³ ÷ 5⁴
   • Step 1: Apply the power-of-a-power rule: (5²)³ = 5^(2×3) = 5⁶.
   • Step 2: Apply the division rule for exponents: 5⁶ ÷ 5⁴ = 5^(6−4) = 5².
   • Step 3: Compute: 5² = 25.

MCQ Traps

• a⁰ = 1 for any nonzero a, including negative bases — commonly missed
• Negative exponent means reciprocal, not a negative number: 2⁻² = 1/4, not −4
• √(a+b) ≠ √a + √b (a very common false simplification)

Self-Test Questions

1. Simplify: (5³ × 5²) ÷ 5⁴
2. Simplify √98 to its simplest radical form.
3. Evaluate 4⁻² and explain why it is not −16.

Past-Paper MCQs (FPSC)

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The FPSC MCQ papers you shared (2024 and 2025) had no direct exponents or roots question (the circle-area MCQ is in Plane Geometry).

Past-Paper Questions with Answers (FPSC)

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Every question below appeared in the FPSC papers you shared; the year and question number are in each title. Answers are worked out by me. The printed PDF garbles the root signs in part (d), so read (b) and (e) as square roots and (c) and (d) as cube roots; check against the original paper.

• CSS 2025 Q8(d): How many prime numbers are between each of the following pairs? (5)

  Part	Range	Primes in it	Count
  (a)	√3 ≈ 1.73 to √120 ≈ 10.95	2, 3, 5, 7	4
  (b)	√10 ≈ 3.16 to √410 ≈ 20.25	5, 7, 11, 13, 17, 19	6
  (c)	∛10 ≈ 2.15 to ∛999 ≈ 9.997	3, 5, 7	3
  (d)	∛28 ≈ 3.04 to √120 ≈ 10.95	5, 7	2
  (e)	√8 ≈ 2.83 to √400 = 20	3, 5, 7, 11, 13, 17, 19	7

  • Method: convert each root to a decimal, then list the primes strictly between the two values

• CSS 2023 Q8(d): y is directly proportional to x², and y = m for a particular x. Find y in terms of m when x is doubled. (5)

  1. y = kx², so m = kx²
  2. With 2x: y = k(2x)² = 4kx² = 4m

CSS Exam Perspective

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• Most repeated areas: estimating roots to find primes in a range (2025), proportion with powers (2023), and power-pattern series
• Tip: memorise squares to 20 and cubes to 10 so you can place roots instantly