Question 1

Simplify: (3^(1/2) × 3^(1/3)) / 3^(1/6)

Accepted Answer

Step 1: Apply product rule in numerator: 3^(1/2) × 3^(1/3) = 3^(1/2 + 1/3) = 3^(3/6 + 2/6) = 3^(5/6). Step 2: Apply quotient rule: 3^(5/6) / 3^(1/6) = 3^(5/6 - 1/6) = 3^(4/6) = 3^(2/3). Step 3: Express as ∛(3²) = ∛9.

Question 2

If √(x + 5) = 7, find x.

Accepted Answer

Step 1: Square both sides: (√(x + 5))² = 7². Step 2: Simplify: x + 5 = 49. Step 3: Solve for x: x = 49 - 5 = 44.

Question 3

Simplify: (2^3)^(2/3) × (2^2)^(1/2)

Accepted Answer

Step 1: Apply power rule (a^m)^n = a^(mn): (2^3)^(2/3) = 2^(3 × 2/3) = 2^2 = 4. Step 2: Simplify second term: (2^2)^(1/2) = 2^(2 × 1/2) = 2^1 = 2. Step 3: Multiply: 4 × 2 = 8.

Question 4

If 2^(x+1) × 2^(x-1) = 256, find the value of x.

Accepted Answer

Step 1: Use the law of indices: 2^(x+1) × 2^(x-1) = 2^((x+1)+(x-1)) = 2^(2x). Step 2: Rewrite 256 as a power of 2: 256 = 2^8. Step 3: Equate exponents: 2^(2x) = 2^8, so 2x = 8, therefore x = 4.

Question 5

Rationalize: 1/(√7 - √5). Express in the form (a√7 + b√5)/c where a, b, c are integers.

Accepted Answer

Step 1: Multiply numerator and denominator by the conjugate (√7 + √5). Step 2: Numerator = 1 × (√7 + √5) = √7 + √5. Step 3: Denominator = (√7 - √5)(√7 + √5) = 7 - 5 = 2. Step 4: Result = (√7 + √5)/2, so a = 1, b = 1, c = 2.

Question 6

Simplify: (∛64 × ⁴√81) / (√16)

Accepted Answer

Step 1: ∛64 = ∛(4³) = 4. Step 2: ⁴√81 = ⁴√(3⁴) = 3. Step 3: √16 = 4. Step 4: (4 × 3) / 4 = 12 / 4 = 3.

Question 7

If (√5 + √3)² - (√5 - √3)² = k√15, find the value of k.

Accepted Answer

Step 1: Use the identity (a + b)² - (a - b)² = 4ab. Step 2: Here, a = √5 and b = √3. Step 3: (√5 + √3)² - (√5 - √3)² = 4 × √5 × √3 = 4√15. Step 4: Therefore, k√15 = 4√15, so k = 4.

Question 8

If x = 3^(1/3) + 3^(-1/3), find the value of x³ - 3x.

Accepted Answer

Step 1: Let a = 3^(1/3) and b = 3^(-1/3). Then x = a + b and ab = 3^(1/3) × 3^(-1/3) = 3⁰ = 1. Step 2: Cube both sides: x³ = (a + b)³ = a³ + b³ + 3ab(a + b) = 3 + 1/3 + 3(1)(x) = 3 + 1/3 + 3x. Step 3: x³ = 10/3 + 3x. Step 4: x³ - 3x = 10/3.

Question 9

If 2^(3x) = 512, find the value of 2^(2x - 1).

Accepted Answer

Step 1: 512 = 2⁹, so 2^(3x) = 2⁹. Step 2: Therefore, 3x = 9, which gives x = 3. Step 3: 2^(2x - 1) = 2^(2(3) - 1) = 2^(6 - 1) = 2⁵ = 32.

Question 10

If x = (√5 + √3)/(√5 - √3) and y = (√5 - √3)/(√5 + √3), find the value of x² + y² + xy.

Accepted Answer

Step 1: Rationalize x. x = (√5 + √3)/(√5 - √3) × (√5 + √3)/(√5 + √3) = (√5 + √3)²/(5 - 3) = (5 + 2√15 + 3)/2 = (8 + 2√15)/2 = 4 + √15. Step 2: Rationalize y. y = (√5 - √3)/(√5 + √3) × (√5 - √3)/(√5 - √3) = (√5 - √3)²/(5 - 3) = (5 - 2√15 + 3)/2 = (8 - 2√15)/2 = 4 - √15. Step 3: Notice y = 1/x (verify: (4 + √15)(4 - √15) = 16 - 15 = 1). Step 4: x + y = 8, xy = 1. Step 5: x² + y² = (x+y)² - 2xy = 64 - 2 = 62. Step 6: x² + y² + xy = 62 + 1 = 63.

Question 11

If 2^(3x-1) × 4^(x+2) = 8^(2x) × 2^(-3), find the value of x.

Accepted Answer

Step 1: Express all terms as powers of 2. 4 = 2², so 4^(x+2) = 2^(2(x+2)) = 2^(2x+4). 8 = 2³, so 8^(2x) = 2^(3×2x) = 2^(6x). Step 2: Rewrite the equation: 2^(3x-1) × 2^(2x+4) = 2^(6x) × 2^(-3). Step 3: Combine exponents on each side: 2^(3x-1+2x+4) = 2^(6x-3), which gives 2^(5x+3) = 2^(6x-3). Step 4: Equate exponents: 5x + 3 = 6x - 3. Step 5: Solve: 3 + 3 = 6x - 5x, so 6 = x.

Question 12

If √(x + √(x + √(x + ...))) = 3, find the value of x.

Accepted Answer

Step 1: Let y = √(x + √(x + √(x + ...))). Since the expression is infinite and self-similar, y = √(x + y). Step 2: We're given that y = 3. Step 3: Substitute: 3 = √(x + 3). Step 4: Square both sides: 9 = x + 3. Step 5: Solve: x = 6.

SSC CPO Surds & Indices

SSC CPO Surds & Indices — Past Exam Questions

Surds & Indices— Rules & Concept

Key Points to Remember

Exam-Specific Tips

60-Second Revision — Surds & Indices