Powers, Indices, and Roots

Powers and roots are the backbone of so many aptitude topics — from simplification to number theory to data interpretation. The laws of exponents let us simplify ugly expressions in seconds, and having common squares and cubes memorized means we can work faster than anyone using a calculator. Let’s get these locked in.

Laws of Exponents

The most common mistakes:

• (a + b)² ≠ a² + b² — Don’t forget the middle term! (a + b)² = a² + 2ab + b²
• (aᵐ)ⁿ ≠ a^(m+n) — It’s a^(m×n). Exponents multiply, not add.
• 0⁰ is undefined (or sometimes treated as 1 by convention, but avoid this in exams)

Squares of 1 to 30

Having these memorized means we can do reverse lookups (square roots) instantly.

Quick Squaring Tricks

Trick 1: Squaring numbers ending in 5

For n5 (a number ending in 5): n5² = n×(n+1) followed by 25

• 25² → 2×3 = 6, append 25 → 625
• 35² → 3×4 = 12, append 25 → 1225
• 75² → 7×8 = 56, append 25 → 5625
• 115² → 11×12 = 132, append 25 → 13225

Trick 2: Squaring numbers near 50

For (50 + x)²: Start with (25 + x), append x²

• 52² → 25 + 2 = 27, append 04 → 2704
• 47² → 25 - 3 = 22, append 09 → 2209

Trick 3: Squaring numbers near 100

For (100 + x)²: Start with (100 + 2x), append x²

• 103² → 100 + 6 = 106, append 09 → 10609
• 97² → 100 - 6 = 94, append 09 → 9409

Cubes of 1 to 15

Fun fact for cube roots: The unit digit of a perfect cube uniquely determines the unit digit of its cube root (unlike squares where both 2² and 8² end in 4/6 patterns). If a cube ends in 8, its cube root ends in 2. If it ends in 2, its root ends in 8. Other digits map to themselves.

Square Root Estimation

When we can’t simplify a square root to a whole number, we can estimate it quickly.

Method: Find the two perfect squares the number falls between, then estimate proportionally.

Example: Estimate √50

• 7² = 49 and 8² = 64
• 50 is just 1 above 49, and the gap from 49 to 64 is 15
• Approximate: 7 + 1/15 ≈ 7.07
• Actual: √50 ≈ 7.071 ✓

Example: Estimate √200

• 14² = 196 and 15² = 225
• 200 is 4 above 196, gap = 29
• Approximate: 14 + 4/29 ≈ 14.14
• Actual: √200 ≈ 14.142 ✓

Powers of 2 and 3 — Worth Memorizing

Worked Examples

Example 1: Simplify: (27)^(2/3) × (16)^(3/4) ÷ (8)^(1/3)

• (27)^(2/3) = (3³)^(2/3) = 3² = 9
• (16)^(3/4) = (2⁴)^(3/4) = 2³ = 8
• (8)^(1/3) = (2³)^(1/3) = 2
• = 9 × 8 ÷ 2 = 36

Example 2: If 2^(x+3) = 4^(x-1), find x.

Convert to same base:

• 2^(x+3) = (2²)^(x-1) = 2^(2x-2)
• So x + 3 = 2x - 2
• x = 5

Example 3: Find the value of: 5⁴ × 5⁻² × 5³ ÷ 5⁴

Using aᵐ × aⁿ = aᵐ⁺ⁿ:

• = 5^(4 + (-2) + 3 - 4)
• = 5^(1)
• = 5

Example 4: What is the square root of 17956?

Step 1: The number ends in 6, so the root ends in 4 or 6. Step 2: 130² = 16900 and 140² = 19600. So root is between 130 and 140. Step 3: Try 134² = 17956. Yes! Answer: 134

(Quick check: 134² = (130 + 4)² = 16900 + 2×130×4 + 16 = 16900 + 1040 + 16 = 17956 ✓)

Example 5: If 9^x - 9^(x-1) = 648, find x.

• 9^x - 9^x × 9⁻¹ = 648
• 9^x (1 - 1/9) = 648
• 9^x × (8/9) = 648
• 9^x = 648 × 9/8 = 729
• 9^x = 729 = 9³
• x = 3

Common Exam Patterns

1. “Simplify expressions with exponents” → Convert everything to same base, then add/subtract powers
2. “If aˣ = bʸ, find relation” → Convert to common base or take logarithm
3. “Find cube root of large number” → Use unit digit and range estimation
4. “Compare: 2³⁰⁰ vs 3²⁰⁰” → Raise both to power 1/100 → compare 2³ vs 3² → 8 vs 9, so 3²⁰⁰ > 2³⁰⁰
5. “a^(m/n) problems” → Rewrite as (ⁿ√a)ᵐ or ⁿ√(aᵐ), whichever is easier to compute

Practice Problems

Q1: If 2ˣ = 32 and 3ʸ = 243, find the value of 2^(x-2) × 3^(y-3).

Q2: Find the cube root of 19683.

Q3: Simplify: (125)^(2/3) + (64)^(2/3) - (27)^(2/3)

Answers:

A1: 2ˣ = 32 = 2⁵, so x = 5. 3ʸ = 243 = 3⁵, so y = 5. Then 2^(5-2) × 3^(5-3) = 2³ × 3² = 8 × 9 = 72

A2: 19683 ends in 3, so cube root ends in 7. 20³ = 8000, 30³ = 27000. Root is between 20-30. Try 27: 27³ = 19683 ✓. Answer: 27

A3: (5³)^(2/3) + (4³)^(2/3) - (3³)^(2/3) = 5² + 4² - 3² = 25 + 16 - 9 = 32