Prelims · 8 Mental Ability · 8.1 Number system and simplification

8.1.8 Surds and indices: laws, rationalising and comparing surds

Bring every term to a common base to solve index questions, rationalise with the conjugate, and compare surds by raising them to a common power.

  • APPSC Group 2 · Prelims
  • Standard level
  • 5 min read
  • Free

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Core facts

  • Laws of indices (a, b > 0):
    • aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ ÷ aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ
    • (ab)ⁿ = aⁿbⁿ; (a/b)ⁿ = aⁿ/bⁿ
    • a⁰ = 1; a⁻ⁿ = 1/aⁿ; a^(m/n) = (ⁿ√a)ᵐ = ⁿ√(aᵐ)
  • If aˣ = aʸ (a > 0, a ≠ 1), then x = y.
  • Surd: an irrational root of a rational number, such as √3 or ∛5. √4 is not a surd.
  • Order of a surd: the root index. ∛5 is a surd of order 3.
  • Like surds have the same irrational part after simplifying: √50 = 5√2 and √18 = 3√2 are like surds; only like surds can be added.
  • Rationalising factor: for √a it is √a; for (√a + √b) it is the conjugate (√a − √b), since (√a + √b)(√a − √b) = a − b.
  • Comparing surds of different orders: raise each to the LCM of the orders.

Example 1. Find 27^(2/3). Solution. (∛27)² = 3². Answer: 9

Example 2. If 2ˣ = 1/32, find x. Solution. 1/32 = 2⁻⁵. Answer: −5

Example 3. Simplify √50 + √18 − √8. Solution. 5√2 + 3√2 − 2√2. Answer: 6√2

Explained

Type 1: exponential equations. Write both sides with the same base, then equate powers.

Example 4. If 9ˣ = 27^(x − 1), find x. (a) 1 (b) 2 (c) 3 (d) 4 Solution. 3^(2x) = 3^(3x − 3), so 2x = 3x − 3. Answer: (c) 3

Example 5. If 5^(x + 3) = 25^(3x − 4), find x. Solution. 5^(x + 3) = 5^(6x − 8), so x + 3 = 6x − 8, 5x = 11. Answer: 11/5

Type 2: simplifying powers.

Example 6. Simplify 16^(0.75) × 64^(−1/3). Solution. 16^(3/4) = 2³ = 8; 64^(−1/3) = 1/4. 8 × 1/4. Answer: 2

Example 7. Simplify (xᵃ/xᵇ)^(a + b) × (xᵇ/xᶜ)^(b + c) × (xᶜ/xᵃ)^(c + a). Solution. Powers: (a − b)(a + b) + (b − c)(b + c) + (c − a)(c + a) = (a² − b²) + (b² − c²) + (c² − a²) = 0. x⁰ = 1. Answer: 1

Type 3: rationalising.

Example 8. Rationalise 6/(√5 + √2). Solution. Multiply by (√5 − √2): 6(√5 − √2)/(5 − 2) = 2(√5 − √2). Answer: 2√5 − 2√2

Type 4: square root of a surd expression. Write a ± 2√b as (√x ± √y)² with x + y = a and xy = b.

Example 9. Find √(11 − 2√30). Solution. Need x + y = 11, xy = 30: x = 6, y = 5. Root = √6 − √5 (positive, since √6 > √5). Answer: √6 − √5

Type 5: comparing surds of different orders.

Example 10. Which is largest: √3, ∛4, ⁴√6? Solution. LCM of 2, 3, 4 is 12. Raise each to the 12th power: 3⁶ = 729, 4⁴ = 256, 6³ = 216. Answer: √3

Speed tips.

  • Convert decimals in indices to fractions at once: 0.75 = 3/4, 0.2 = 1/5, 1.5 = 3/2.
  • In Type 4, the number under the inner root is the product xy, and the whole number is the sum x + y.

Common traps

  • ✗ "a⁰ = 0." ✓ a⁰ = 1 for any non-zero a.
  • ✗ "aᵐ × bᵐ = (ab)²ᵐ." ✓ It is (ab)ᵐ; powers add only for the same base.
  • ✗ "√5 + √3 = √8." ✓ Only like surds add; √5 + √3 cannot be simplified.
  • ✗ "√(11 − 2√30) = √5 − √6." ✓ A square root is positive: √6 − √5.
  • ✗ "⁴√6 > √3 because 6 > 3." ✓ Compare at a common power: 6³ = 216 < 3⁶ = 729, so √3 is larger.
  • ✗ "(2³)² = 2⁹." ✓ (aᵐ)ⁿ = aᵐⁿ = 2⁶; 2^(3²) = 2⁹ is a different expression.

Quick revision

  • aᵐaⁿ = aᵐ⁺ⁿ; (aᵐ)ⁿ = aᵐⁿ; a⁻ⁿ = 1/aⁿ; a^(m/n) = (ⁿ√a)ᵐ.
  • Same base, equal powers: equate the exponents.
  • Like surds only can be added or subtracted.
  • Conjugate of √a + √b is √a − √b; product a − b.
  • √(a ± 2√b) = √x ± √y with x + y = a, xy = b.
  • Compare surds at the LCM of their orders.
  • Compare √a + √b sums by squaring.
  • 2ᵃ = 3ᵇ = 6ᶜ ⇒ 1/a + 1/b = 1/c.
  • 0.75 = 3/4; 0.25 = 1/4; 1.5 = 3/2 in indices.
Go deeper: Deeper (beyond what APPSC Group 2 usually asks)

Deeper Advanced

Example 11. If 4ˣ − 4^(x − 1) = 24, find x. (a) 3/2 (b) 2 (c) 5/2 (d) 3 Solution. Factor 4^(x − 1): 4^(x − 1)(4 − 1) = 24, so 4^(x − 1) = 8. 2^(2x − 2) = 2³, so 2x − 2 = 3. Answer: (c) 5/2

Example 12. If 2ᵃ = 3ᵇ = 6ᶜ, express c in terms of a and b. Solution. Let the common value be k. Then 2 = k^(1/a), 3 = k^(1/b), 6 = k^(1/c). Since 2 × 3 = 6: 1/a + 1/b = 1/c. Answer: c = ab/(a + b)

Example 13. Find (√5 + √3)/(√5 − √3) + (√5 − √3)/(√5 + √3). Solution. The first term rationalises to (8 + 2√15)/2 = 4 + √15, the second to 4 − √15. The surds cancel. Answer: 8

Example 14. Which is larger: √7 + √3 or √6 + 2? Solution. Both are positive; compare squares. (√7 + √3)² = 10 + 2√21; (√6 + √4)² = 10 + 2√24. √24 > √21. Answer: √6 + 2

  • Rule: for equal sums (7 + 3 = 6 + 4), the pair whose parts are closer together gives the larger sum of roots.
  • Negative and fractional powers together: (0.25)^(−3/2) = 4^(3/2) = 8; 125^(−2/3) = 1/25.
  • Mixed bases: 32^(1/5) × 81^(1/4) = 2 × 3 = 6.

Sources: NCERT Mathematics textbooks, Class 8 (exponents and powers) and Class 9 (number systems: laws of exponents for real numbers, rationalising the denominator)

Last checked 2026-10-17.

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