Question 1

Simplify: (2³ × 2⁵) ÷ 2⁶

Accepted Answer

Step 1: Using the law of indices, when multiplying powers with the same base, add the exponents: 2³ × 2⁵ = 2^(3+5) = 2⁸. Step 2: When dividing powers with the same base, subtract the exponents: 2⁸ ÷ 2⁶ = 2^(8-6) = 2². Step 3: 2² = 4. Therefore, the answer is 4.

Question 2

Evaluate: ∛(125) + √(64)

Accepted Answer

Step 1: Find ∛(125). Since 5³ = 125, ∛(125) = 5. Step 2: Find √(64). Since 8² = 64, √(64) = 8. Step 3: Add them: 5 + 8 = 13.

Question 3

Simplify: (2³)² ÷ 2⁴

Accepted Answer

Step 1: Apply power rule (aᵐ)ⁿ = aᵐⁿ. So (2³)² = 2⁶. Step 2: Apply quotient rule aᵐ ÷ aⁿ = aᵐ⁻ⁿ. So 2⁶ ÷ 2⁴ = 2⁶⁻⁴ = 2². Step 3: Calculate 2² = 4.

Question 4

Simplify: (5²)³ ÷ 5⁴

Accepted Answer

Step 1: Using the power rule (a^m)^n = a^(m×n), we have (5²)³ = 5^(2×3) = 5⁶. Step 2: Now divide: 5⁶ ÷ 5⁴ = 5^(6-4) = 5². Step 3: 5² = 25. Therefore, the answer is 25.

Question 5

Simplify: √(50) + √(8) − √(18)

Accepted Answer

Step 1: Express each surd in simplest form. √(50) = √(25 × 2) = 5√2. √(8) = √(4 × 2) = 2√2. √(18) = √(9 × 2) = 3√2. Step 2: Combine like surds: 5√2 + 2√2 − 3√2 = (5 + 2 − 3)√2 = 4√2.

Question 6

If 3^x = 243, find the value of x.

Accepted Answer

Step 1: Express 243 as a power of 3. We know: 3¹ = 3, 3² = 9, 3³ = 27, 3⁴ = 81, 3⁵ = 243. Step 2: Therefore, 3^x = 3⁵. Step 3: Comparing both sides, x = 5. Therefore, the answer is 5.

Question 7

If 3ˣ = 81, find the value of x.

Accepted Answer

Step 1: Express 81 as a power of 3. 81 = 3⁴ (since 3 × 3 × 3 × 3 = 81). Step 2: Rewrite the equation: 3ˣ = 3⁴. Step 3: Since the bases are equal, the exponents must be equal: x = 4.

Question 8

Rationalize: 1/(√5 − √3)

Accepted Answer

Step 1: Multiply numerator and denominator by the conjugate (√5 + √3). Step 2: Numerator becomes 1 × (√5 + √3) = √5 + √3. Step 3: Denominator becomes (√5 − √3)(√5 + √3) = (√5)² − (√3)² = 5 − 3 = 2. Step 4: Result is (√5 + √3)/2.

Question 9

Simplify: (16)^(1/4) × (8)^(1/3)

Accepted Answer

Step 1: Evaluate (16)^(1/4). Since 16 = 2⁴, (16)^(1/4) = (2⁴)^(1/4) = 2^(4/4) = 2¹ = 2. Step 2: Evaluate (8)^(1/3). Since 8 = 2³, (8)^(1/3) = (2³)^(1/3) = 2^(3/3) = 2¹ = 2. Step 3: Multiply: 2 × 2 = 4.

Question 10

Simplify: (5^(1/2) + 3^(1/2))² - (5^(1/2) - 3^(1/2))²

Accepted Answer

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

Question 11

If (x^(1/3))^2 = 4, find x.

Accepted Answer

Step 1: (x^(1/3))^2 = x^(2/3) = 4. Step 2: Raise both sides to the power 3/2: (x^(2/3))^(3/2) = 4^(3/2). Step 3: x^(2/3 × 3/2) = 4^(3/2). Step 4: x¹ = (2²)^(3/2) = 2³ = 8. Therefore x = 8.

Question 12

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

Accepted Answer

Step 1: Use the rule a^m × a^n = a^(m+n) for the numerator: 3^(1/2) × 3^(1/3) = 3^(1/2 + 1/3) = 3^(3/6 + 2/6) = 3^(5/6). Step 2: Use the rule a^m / a^n = a^(m−n): 3^(5/6) / 3^(1/6) = 3^(5/6 − 1/6) = 3^(4/6) = 3^(2/3). Therefore, answer is 3^(2/3) or ∛9.

Question 13

If √(2 + √3) = √a + √b, where a and b are rational numbers, then find a + b.

Accepted Answer

Step 1: Square both sides: 2 + √3 = (√a + √b)² = a + b + 2√(ab). Step 2: Equate rational and irrational parts: a + b = 2 and 2√(ab) = √3. Step 3: From 2√(ab) = √3, square both sides: 4ab = 3, so ab = 3/4. Step 4: We have a + b = 2 and ab = 3/4. These are roots of t² − 2t + 3/4 = 0. Step 5: Using the quadratic formula: t = (2 ± √(4 − 3))/2 = (2 ± 1)/2. So t = 3/2 or t = 1/2. Thus a = 3/2, b = 1/2 (or vice versa). Step 6: a + b = 3/2 + 1/2 = 2. Therefore, answer is 2.

Question 14

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 15

If 3^x = 27 and 2^y = 32, find x + y.

Accepted Answer

Step 1: 3^x = 27 = 3³, so x = 3. Step 2: 2^y = 32 = 2⁵, so y = 5. Step 3: x + y = 3 + 5 = 8.

Question 16

If 2^(x+1) × 4^(x-1) = 64, find the value of x.

Accepted Answer

Step 1: Rewrite 4 and 64 as powers of 2: 4 = 2² and 64 = 2⁶. Step 2: 2^(x+1) × (2²)^(x-1) = 2⁶. Step 3: 2^(x+1) × 2^(2x-2) = 2⁶. Step 4: 2^(x+1+2x-2) = 2⁶. Step 5: 2^(3x-1) = 2⁶. Step 6: 3x - 1 = 6, so 3x = 7, thus x = 7/3.

Question 17

If √(x + √(x + √(x + ...))) = 5, 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 y = 5, so 5 = √(x + 5). Step 3: Square both sides: 25 = x + 5. Step 4: Solve for x: x = 20.

Question 18

If 5^(2x) - 5^x - 12 = 0, find the value of 5^x.

Accepted Answer

Step 1: Let y = 5^x. Then the equation becomes y² - y - 12 = 0. Step 2: Factor: (y - 4)(y + 3) = 0. Step 3: Solve: y = 4 or y = -3. Step 4: Since 5^x must be positive, 5^x = 4.

Question 19

If 3^x = 27^(y-1) and 2^(2x) = 16^y, find the value of x + y.

Accepted Answer

Step 1: Simplify the first equation. 3^x = 27^(y-1) = (3^3)^(y-1) = 3^(3y-3). Therefore, x = 3y - 3. Step 2: Simplify the second equation. 2^(2x) = 16^y = (2^4)^y = 2^(4y). Therefore, 2x = 4y, so x = 2y. Step 3: Solve the system. From x = 3y - 3 and x = 2y: 2y = 3y - 3 → -y = -3 → y = 3. Step 4: Substitute back: x = 2(3) = 6. Therefore, x + y = 6 + 3 = 9.

Question 20

Simplify: (∛(64) + ∛(125))² - (∛(27) - ∛(8))² and express the answer in simplest form.

Accepted Answer

Step 1: Calculate the cube roots: ∛(64) = 4, ∛(125) = 5, ∛(27) = 3, ∛(8) = 2. Step 2: Substitute: (4 + 5)² - (3 - 2)² = 9² - 1² = 81 - 1 = 80.

SSC CHSL Surds & Indices

Surds & Indices— Rules & Concept

Key Points to Remember

Exam-Specific Tips

Surds & Indices — Practice Questions

60-Second Revision — Surds & Indices