Question 1

What is the value of (125)^(2/3) × (32)^(-2/5) ÷ (64)^(1/2)?

Accepted Answer

Step 1: (125)^(2/3) = (5³)^(2/3) = 5² = 25. Step 2: (32)^(-2/5) = (2⁵)^(-2/5) = 2^(-2) = 1/4. Step 3: (64)^(1/2) = 8. Step 4: Expression = 25 × (1/4) ÷ 8 = 25/(4×8) = 25/32. Therefore, the answer is 25/32.

Question 2

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

Accepted Answer

Step 1: Solve 3^x = 27. Express 27 as a power of 3: 27 = 3³, so x = 3. Step 2: Solve 2^y = 32. Express 32 as a power of 2: 32 = 2⁵, so y = 5. Step 3: Add: x + y = 3 + 5 = 8.

Question 3

Simplify: (16^(3/4) − 8^(2/3)) / (4^(1/2))

Accepted Answer

Step 1: Calculate 16^(3/4) = (2⁴)^(3/4) = 2³ = 8. Step 2: Calculate 8^(2/3) = (2³)^(2/3) = 2² = 4. Step 3: Calculate numerator: 8 − 4 = 4. Step 4: Calculate 4^(1/2) = 2. Step 5: Divide: 4 / 2 = 2.

Question 4

Simplify: (5^(2/3) × 5^(1/3)) / 5^(−1/3)

Accepted Answer

Step 1: Apply product rule in numerator: 5^(2/3) × 5^(1/3) = 5^(2/3 + 1/3) = 5¹ = 5. Step 2: Apply quotient rule: 5¹ / 5^(−1/3) = 5^(1 − (−1/3)) = 5^(1 + 1/3) = 5^(4/3). Step 3: Express as radical: 5^(4/3) = ∛(5⁴) = ∛625.

Question 5

Simplify: (√5 + √3)² + (√5 - √3)² - 2(√5 + √3)(√5 - √3).

Accepted Answer

Step 1: Let x = √5 + √3 and y = √5 - √3. The expression becomes x² + y² - 2xy = (x - y)². Step 2: Calculate x - y = (√5 + √3) - (√5 - √3) = 2√3. Step 3: (2√3)² = 4 × 3 = 12.

Question 6

If 3^x = 5^y = 15^z, then which of the following is true?

Accepted Answer

Step 1: Let 3^x = 5^y = 15^z = k (some constant). Step 2: From 3^x = k, we get x = log₃(k). From 5^y = k, we get y = log₅(k). From 15^z = k, we get z = log₁₅(k). Step 3: Using change of base: 1/x = log(3)/log(k), 1/y = log(5)/log(k), 1/z = log(15)/log(k). Step 4: Since 15 = 3 × 5, we have log(15) = log(3) + log(5). Step 5: Therefore, 1/z = (log(3) + log(5))/log(k) = 1/x + 1/y. This gives 1/z = 1/x + 1/y.

Question 7

Simplify: (∛64 + ∛27)² - (∛64 - ∛27)² / (∛64 × ∛27).

Accepted Answer

Step 1: ∛64 = 4 and ∛27 = 3. Step 2: Numerator = (4 + 3)² - (4 - 3)² = 7² - 1² = 49 - 1 = 48. Step 3: Denominator = 4 × 3 = 12. Step 4: Result = 48/12 = 4.

Question 8

If x = (2 + √3)^(1/3) + (2 - √3)^(1/3), then find x³ - 3x.

Accepted Answer

Step 1: Let a = (2 + √3)^(1/3) and b = (2 - √3)^(1/3), so x = a + b. Step 2: Calculate ab = [(2 + √3)(2 - √3)]^(1/3) = [4 - 3]^(1/3) = 1^(1/3) = 1. Step 3: Cube both sides: x³ = (a + b)³ = a³ + b³ + 3ab(a + b) = (2 + √3) + (2 - √3) + 3(1)(x) = 4 + 3x. Step 4: Therefore, x³ - 3x = 4.

Question 9

If (2^a × 3^b × 5^c)^(1/6) = 2^(1/2) × 3^(1/3) × 5^(1/6), then find the value of (a + b + c).

Accepted Answer

Step 1: Expand the left side: 2^(a/6) × 3^(b/6) × 5^(c/6) = 2^(1/2) × 3^(1/3) × 5^(1/6). Step 2: Compare exponents of like bases. For base 2: a/6 = 1/2, so a = 3. For base 3: b/6 = 1/3, so b = 2. For base 5: c/6 = 1/6, so c = 1. Step 3: a + b + c = 3 + 2 + 1 = 6.

Question 10

Simplify: (√8 + √2) ÷ √2

Accepted Answer

Step 1: Simplify √8 = √(4×2) = 2√2. Step 2: Substitute: (2√2 + √2) ÷ √2 = 3√2 ÷ √2. Step 3: Divide: 3√2 ÷ √2 = 3.

Question 11

Simplify: (5⁻²) × (5³) / (5⁰)

Accepted Answer

Step 1: Apply exponent rules. 5⁻² × 5³ = 5⁻²⁺³ = 5¹ = 5 (using aᵐ × aⁿ = aᵐ⁺ⁿ). Step 2: Divide by 5⁰. Since 5⁰ = 1, we have 5 / 1 = 5.

Question 12

Rationalize the denominator: 5/(√5)

Accepted Answer

Step 1: Multiply numerator and denominator by √5. [5 × √5] / [√5 × √5]. Step 2: Simplify numerator: 5√5. Step 3: Simplify denominator: (√5)² = 5. Step 4: Result: 5√5 / 5 = √5.

Question 13

Simplify: (2^5 × 2^3) ÷ 2^6

Accepted Answer

Step 1: Using the law of indices, a^m × a^n = a^(m+n), we get 2^5 × 2^3 = 2^(5+3) = 2^8. Step 2: Using a^m ÷ a^n = a^(m-n), we get 2^8 ÷ 2^6 = 2^(8-6) = 2^2 = 4.

Question 14

Simplify: √(144) + √(64) − √(36)

Accepted Answer

Step 1: √144 = 12. Step 2: √64 = 8. Step 3: √36 = 6. Step 4: 12 + 8 − 6 = 14.

Question 15

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: Set up equation: 3ˣ = 3⁴. Step 3: Since bases are equal, exponents must be equal: x = 4.

Question 16

If 5ˣ = 125, then find the value of 5^(x−1).

Accepted Answer

Step 1: Express 125 as a power of 5: 125 = 5³. Step 2: Since 5ˣ = 5³, we have x = 3. Step 3: Calculate 5^(x−1) = 5^(3−1) = 5² = 25. Therefore, the answer is 25.

Question 17

Simplify: (2^3)^2 × 2^(-4)

Accepted Answer

Step 1: Using (a^m)^n = a^(mn), we get (2^3)^2 = 2^6. Step 2: Using a^m × a^n = a^(m+n), we get 2^6 × 2^(-4) = 2^(6-4) = 2^2 = 4.

Question 18

Simplify: √(48) + √(12) − √(27)

Accepted Answer

Step 1: Express each surd in simplest form. √(48) = √(16 × 3) = 4√3. Step 2: √(12) = √(4 × 3) = 2√3. Step 3: √(27) = √(9 × 3) = 3√3. Step 4: Combine like terms: 4√3 + 2√3 − 3√3 = (4 + 2 − 3)√3 = 3√3.

Question 19

Evaluate: (3⁻²)²

Accepted Answer

Step 1: Using the power rule (aᵐ)ⁿ = aᵐⁿ, we have (3⁻²)² = 3^(-2×2) = 3⁻⁴. Step 2: Convert negative exponent to positive: 3⁻⁴ = 1/3⁴. Step 3: Calculate 3⁴ = 81, so 3⁻⁴ = 1/81. Therefore, the answer is 1/81.

Question 20

Evaluate: ∛(125) + ∜(16)

Accepted Answer

Step 1: Find ∛(125). Since 5³ = 125, ∛(125) = 5. Step 2: Find ∜(16). Since 2⁴ = 16, ∜(16) = 2. Step 3: Add: 5 + 2 = 7.

Question 21

Simplify: (2³)² ÷ 2⁴

Accepted Answer

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

Question 22

Evaluate: (3^4)^2 ÷ 3^5

Accepted Answer

Step 1: Using the power rule (a^m)^n = a^(mn), we get (3^4)^2 = 3^(4×2) = 3^8. Step 2: Using a^m ÷ a^n = a^(m-n), we get 3^8 ÷ 3^5 = 3^(8-5) = 3^3 = 27.

Question 23

Simplify: √(144) + ∛(27) − ⁴√(16)

Accepted Answer

Step 1: Calculate √144 = 12 (since 12² = 144). Step 2: Calculate ∛27 = 3 (since 3³ = 27). Step 3: Calculate ⁴√16 = 2 (since 2⁴ = 16). Step 4: Combine: 12 + 3 − 2 = 13. Therefore, the answer is 13.

Question 24

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: Calculate 2² = 4. Therefore, the answer is 4.

Question 25

If 5^x = 125, find the value of x.

Accepted Answer

Step 1: Express 125 as a power of 5: 125 = 5^3. Step 2: Since 5^x = 5^3, we have x = 3.

Question 26

Simplify: (∛64)² + (∜81)² − (√16)³

Accepted Answer

Step 1: Calculate (∛64)² = (4)² = 16. Step 2: Calculate (∜81)² = (3)² = 9. Step 3: Calculate (√16)³ = (4)³ = 64. Step 4: Combine: 16 + 9 − 64 = −39.

Question 27

If 2^(3x−1) = 128, find the value of x.

Accepted Answer

Step 1: Express 128 as a power of 2: 128 = 2⁷. Step 2: Rewrite the equation: 2^(3x−1) = 2⁷. Step 3: Equate exponents: 3x − 1 = 7. Step 4: Solve: 3x = 8, so x = 8/3.

Question 28

Simplify: (16)^(−3/4) × (8)^(2/3) ÷ (32)^(−1/5)

Accepted Answer

Step 1: Convert to prime bases. 16 = 2⁴, 8 = 2³, 32 = 2⁵. Step 2: (2⁴)^(−3/4) = 2^(−3), (2³)^(2/3) = 2², (2⁵)^(−1/5) = 2^(−1). Step 3: 2^(−3) × 2² ÷ 2^(−1) = 2^(−3+2−(−1)) = 2^(−3+2+1) = 2⁰ = 1.

Question 29

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

Accepted Answer

Step 1: Use the law 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 law 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 ∛(3²) or ∛9.

RRB NTPC Surds & Indices

RRB NTPC Surds & Indices — Past Exam Questions

Surds & Indices— Rules & Concept

Key Points to Remember

Exam-Specific Tips

Surds & Indices — Practice Questions

60-Second Revision — Surds & Indices