Square roots, cube roots, surds and indices
A square root undoes squaring and a cube root undoes cubing. Laws of indices add, subtract or multiply powers; surds are roots that stay irrational.
Laws of indices and surds
| Law | Rule | Example |
|---|---|---|
| Product | a^m × a^n = a^(m+n) | 2³ × 2⁴ = 2⁷ = 128 |
| Quotient | a^m ÷ a^n = a^(m−n) | 5⁶ ÷ 5⁴ = 5² = 25 |
| Power of a power | (a^m)^n = a^(mn) | (3²)³ = 3⁶ = 729 |
| Zero power | a⁰ = 1 (a ≠ 0) | 7⁰ = 1 |
| Negative power | a^(−n) = 1/a^n | 2^(−3) = 1/8 |
| Fractional power | a^(m/n) = (nth root of a)^m | 8^(2/3) = 2² = 4 |
| Surd product | √a × √b = √(ab) | √12 = √4 × √3 = 2√3 |
| Rationalising | Multiply top and bottom by the conjugate | 1/(√3 + √2) = √3 − √2 |
| Perfect squares end in | 0, 1, 4, 5, 6 or 9 only | a number ending in 7 is never a perfect square |
Worked example 1: square root
Find √7,056.
- Standard method (prime factors): 7,056 = 2⁴ × 3² × 7². Halve each power: √7,056 = 2² × 3 × 7 = 84.
- Shortcut (unit digit and range): 7,056 ends in 6, so the root ends in 4 or 6. 80² = 6,400 and 90² = 8,100, so the root is 84 or 86. 85² = 7,225 is more than 7,056, so the root is below 85: it is 84.
- Check: 84 × 84 = 7,056.
84.
Worked example 2: cube root
Find the cube root of 17,576.
- Standard method (prime factors): 17,576 = 2³ × 13³, so the cube root is 2 × 13 = 26.
- Shortcut (last digit and leading group): the number ends in 6, and a cube ending in 6 has a root ending in 6 (endings 0, 1, 4, 5, 6, 9 stay the same; 2 and 8 swap; 3 and 7 swap). Strike off the last three digits: 17 lies between 2³ = 8 and 3³ = 27, so the tens digit is 2. Root = 26.
- Check: 26 × 26 = 676 and 676 × 26 = 17,576.
26.
Worked example 3: laws of indices
Simplify 27^(2/3) × 16^(−3/4) ÷ 81^(1/4).
- Standard method (term by term): 27^(2/3) = (cube root of 27)² = 3² = 9; 16^(−3/4) = 1 ÷ (fourth root of 16)³ = 1 ÷ 2³ = 1/8; 81^(1/4) = 3.
- So the value is 9 × 1/8 ÷ 3 = 9/24 = 3/8.
- Shortcut (prime bases): 27^(2/3) = 3², 16^(−3/4) = 2^(−3), 81^(1/4) = 3¹. Combine: 3^(2−1) × 2^(−3) = 3/8.
- Check: 9 × 0.125 = 1.125, and 1.125 ÷ 3 = 0.375 = 3/8.
3/8.
Worked example 4: comparing surds
Arrange √2, ∛3 and ∜5 in ascending order.
- Standard method (common power): the root orders are 2, 3 and 4, with LCM 12. Raise each to the 12th power: (√2)¹² = 2⁶ = 64; (∛3)¹² = 3⁴ = 81; (∜5)¹² = 5³ = 125.
- 64 < 81 < 125, so √2 < ∛3 < ∜5.
- Shortcut (decimals): √2 ≈ 1.414; ∛3 ≈ 1.442 (since 1.44³ ≈ 2.99); ∜5 ≈ 1.495 (since 1.5⁴ ≈ 5.06).
- Check: both methods give the same order.
√2 < ∛3 < ∜5.
SSC and railway papers ask square roots, cube roots and surd simplification as quick one-liners; bank papers hide them inside simplification sets. Placement tests like 'find x' with indices, such as 2^(x+3) = 32^(x−1).
2 marks: “Find the value of 64^(2/3).” 64^(2/3) = (cube root of 64)² = 4² = 16.
5 marks: “State any four laws of indices with an example each.” Product, quotient, power of a power, zero, negative and fractional powers, each with a short numerical example such as 2³ × 2⁴ = 2⁷.
10 marks: “(a) Find √5,184 by prime factors. (b) Find the cube root of 9,261. (c) Simplify 27^(2/3) × 16^(−3/4) ÷ 81^(1/4). (d) Rationalise 1/(√5 − √3).” (a) 5,184 = 2⁶ × 3⁴, root 72. (b) 9,261 = 3³ × 7³, root 21. (c) 3/8. (d) (√5 + √3)/2. Show a check for each.
Same base: multiply → add powers; divide → subtract; power of a power → multiply. a⁰ = 1, a^(−n) = 1/a^n. Squares end in 0, 1, 4, 5, 6, 9, never 2, 3, 7 or 8.