Surds & Indices
Key Concepts
| # | Concept | Explanation |
|---|---|---|
| 1 | Surds | Irrational roots that cannot be simplified to a whole number (e.g., √2, ³√5). |
| 2 | Indices | Powers or exponents indicating how many times a number is multiplied by itself. |
| 3 | Rationalisation | Process of eliminating a surd from the denominator by multiplying numerator and denominator by the conjugate. |
| 4 | Laws of Indices | Rules: aᵐ × aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁻ⁿ = 1/aⁿ, a⁰ = 1. |
| 5 | Comparing Surds | Convert to the same order (LCM of roots) or decimal approximation to compare. |
| 6 | Simplifying Surds | Factor the number inside the root into perfect-square/cube factors and pull them out. |
| 7 | Double Indices | Expression like (aᵐ)ⁿ simplifies to aᵐⁿ; handle bracket first. |
| 8 | Mixed Operations | BODMAS still applies—simplify brackets, then indices, then multiplication/division. |
15 Practice MCQs
- (64)1/2 + (27)1/3 = ?
Options
A. 5
B. 7
C. 11
D. 14
Answer: C
Solution: √64 = 8; ³√27 = 3 → 8 + 3 = 11
Shortcut: Memorise perfect squares up to 30 and cubes up to 15.
Tag: Basic indices + surds
- (25 × 23) ÷ 26 = ?
Options
A. 2
B. 4
C. 8
D. 16
Answer: B
Solution: 25+3-6 = 22 = 4
Shortcut: Add/subtract exponents directly when bases are same.
Tag: Laws of indices
- The value of (0.04)-1/2 is
Options
A. 0.2
B. 5
C. 25
D. 1/5
Answer: B
Solution: (4/100)-1/2 = (100/4)1/2 = √25 = 5
Shortcut: Flip fraction when exponent is negative.
Tag: Negative index
- Simplify: 5√3 - 2√12 + √75
Options
A. 4√3
B. 6√3
C. 8√3
D. 10√3
Answer: B
Solution: √12 = 2√3; √75 = 5√3 → 5√3 - 4√3 + 5√3 = 6√3
Shortcut: Break surds into simplest form first.
Tag: Surd simplification
- If 3x = 81, then x = ?
Options
A. 3
B. 4
C. 5
D. 6
Answer: B
Solution: 81 = 34 ⇒ x = 4
Shortcut: Express RHS as power of same base.
Tag: Exponential equation
- Rationalise: 1/(√7 + √2)
Options
A. (√7 - √2)/5
B. (√7 + √2)/5
C. (√7 - √2)/3
D. (√7 + √2)/9
Answer: A
Solution: Multiply by (√7 - √2)/(√7 - √2) → (7 - 2)/(7 - 2) = 5 → numerator = √7 - √2
Shortcut: Use (a+b)(a-b) = a²-b².
Tag: Rationalisation
- (16)3/4 × (8)2/3 = ?
Options
A. 16
B. 24
C. 32
D. 48
Answer: C
Solution: 163/4 = (24)3/4 = 23 = 8; 82/3 = 22 = 4 → 8 × 4 = 32
Shortcut: Convert everything to same prime base (2).
Tag: Fractional indices
- Which is greatest? √3, ³√4, ⁴√5
Options
A. √3
B. ³√4
C. ⁴√5
D. All equal
Answer: A
Solution: Raise each to 12th power (LCM of 2,3,4): 36=729; 44=256; 53=125 → 729 largest
Shortcut: LCM power comparison.
Tag: Comparing surds
- (50 + 70) ÷ 20 = ?
Options
A. 0
B. 1
C. 2
D. Undefined
Answer: C
Solution: 1 + 1 = 2; 2 ÷ 1 = 2
Shortcut: Anything to power 0 is 1.
Tag: Zero index
- If √x = 0.25, then x = ?
Options
A. 0.5
B. 0.0625
C. 0.125
D. 0.025
Answer: B
Solution: x = (0.25)² = 0.0625
Shortcut: Square both sides instantly.
Tag: Square root equation
- Simplify: (2√5)2
Options
A. 10
B. 20
C. 40
D. 100
Answer: B
Solution: 22 × (√5)2 = 4 × 5 = 20
Shortcut: Square coefficient and surd separately.
Tag: Surd squaring
- (0.2)3 × (0.04)-2 = ?
Options
A. 5
B. 25
C. 125
D. 625
Answer: C
Solution: (1/5)3 × (1/25)-2 = 1/125 × 625 = 5 → 625/125 = 5 (Oops!)
Correction: (0.04)-2 = (25)2 = 625; (0.2)3 = 0.008 → 0.008 × 625 = 5
Answer: A
Shortcut: Convert decimals to fractions first.
Tag: Negative index
- ³√0.000001 = ?
Options
A. 0.01
B. 0.001
C. 0.0001
D. 0.1
Answer: A
Solution: 0.000001 = 10-6 → (10-6)1/3 = 10-2 = 0.01
Shortcut: Recognise 10-6 as (10-2)3.
Tag: Cube root
- If 2x-1 + 2x+1 = 160, then x = ?
Options
A. 5
B. 6
C. 7
D. 8
Answer: B
Solution: 2x-1(1 + 4) = 160 → 5·2x-1 = 160 → 2x-1 = 32 → x-1 = 5 → x = 6
Shortcut: Factor common smaller exponent.
Tag: Exponential equation
- The rationalising factor of ³√5 is
Options
A. ³√5
B. ³√25
C. ³√125
D. ³√1
Answer: B
Solution: ³√5 × ³√25 = ³√125 = 5 (rational)
Shortcut: Need exponent sum to 3 (order of root).
Tag: Rationalising factor
Speed Tricks
| Situation | Shortcut | Example |
|---|---|---|
| Comparing surds | Raise to LCM power | ³√4 vs √3 → 12th power → 44=256 vs 36=729 → √3 wins |
| Decimal negative index | Flip & positive | (0.04)-1/2 → (100/4)1/2 = 5 |
| 0.1, 0.01, 0.001 powers | Write as 10⁻ⁿ | (0.001)1/3 = (10⁻³)1/3 = 10⁻¹ = 0.1 |
| Sum of exponentials | Factor smallest term | 3x + 3x+2 = 3x(1+9) = 10·3x |
| Last digit of power | Cycle last digit | 783 → 7,9,3,1 cycle → 83 mod 4 = 3 → last digit 3 |
Quick Revision
| Point | Detail |
|---|---|
| 1 | √a × √a = a; √a × √b = √(ab) |
| 2 | aᵐ × aⁿ = aᵐ⁺ⁿ; never add exponents of different bases |
| 3 | (aᵐ)ⁿ = aᵐⁿ; power of power → multiply exponents |
| 4 | a⁻ⁿ = 1/aⁿ; flip and sign changes |
| 5 | a⁰ = 1 for any a ≠ 0 |
| 6 | To compare surds, bring to same root order (LCM) |
| 7 | Rationalise denominators using conjugate (a±√b) |
| 8 | Simplify surds by pulling out perfect-square factors |
| 9 | ³√a × ³√a² = a (rationalising factor pair) |
| 10 | BODMAS rule still governs—brackets before indices |
BETA
