Example 1 · level 1
If (7²)⁶ ÷ 7⁶ = 7ˣ, find x.
- 2
- 6
- 18
- 58
Show solution
- (7²)⁶ = 7¹².
- 7¹² ÷ 7⁶ = 7⁶.
- So x = 6.
Answer: 6
ABHYAS PREP: ADVANCED QUANT · SURDS & INDICES
Laws of indices and surds
Same base: multiply → add powers, divide → subtract, power of a power → multiply. a⁻ⁿ = 1/aⁿ and a^(p/q) = (q-th root of a)ᵖ. Surds combine only when the number under the root is the same.
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Worked examples
Example 2 · level 2
If 32^(x + 2) = 2¹⁵, find x.
- 1
- 13/5
- 3
- 13
Show solution
- 32^(x + 2) = 2^(5x + 10).
- 5x + 10 = 15 ⇒ 5x = 5.
- x = 1.
Answer: 1
Example 3 · level 3
Rationalise the denominator and simplify: 28/(√14 + √7).
- 28(√14 − √7)/21
- 4(√14 + √7)
- 4(√14 − √7)
- 28(√14 − √7)
Show solution
- 28/(√14 + √7) × (√14 − √7)/(√14 − √7).
- Denominator: 14 − 7 = 7. Numerator: 28(√14 − √7).
- 28/7 = 4, so the answer is 4(√14 − √7).
Answer: 4(√14 − √7)