Question 1

Simplify: ∛(27) + ∜(16) − √(4)

Accepted Answer

Step 1: ∛(27) = ∛(3³) = 3. Step 2: ∜(16) = ∜(2⁴) = 2. Step 3: √(4) = 2. Step 4: Add and subtract: 3 + 2 − 2 = 3.

Question 2

Rationalize: 1/(√7 − √2)

Accepted Answer

Step 1: Multiply numerator and denominator by the conjugate (√7 + √2). Step 2: Numerator becomes √7 + √2. Step 3: Denominator becomes (√7)² − (√2)² = 7 − 2 = 5. Step 4: Result is (√7 + √2)/5.

Question 3

If 2^(x+2) = 64, find the value of x.

Accepted Answer

Step 1: Rewrite 64 as a power of 2: 64 = 2⁶. Step 2: Set exponents equal: 2^(x+2) = 2⁶, so x + 2 = 6. Step 3: Solve: x = 4.

Question 4

Simplify: (3⁴ × 3⁻²) ÷ 3¹

Accepted Answer

Step 1: Apply product rule: 3⁴ × 3⁻² = 3^(4−2) = 3². Step 2: Apply quotient rule: 3² ÷ 3¹ = 3^(2−1) = 3¹ = 3.

Question 5

If 5^(2x) = 125, find x.

Accepted Answer

Step 1: Express 125 as a power of 5: 125 = 5³. Step 2: Set exponents equal: 5^(2x) = 5³, so 2x = 3. Step 3: Solve: x = 3/2 = 1.5.

Question 6

If 5^(2x) = 125, then find the value of 5^(x−1).

Accepted Answer

Step 1: Rewrite 125 as a power of 5: 125 = 5³. Step 2: So 5^(2x) = 5³, which means 2x = 3, thus x = 3/2. Step 3: Calculate 5^(x−1) = 5^(3/2 − 1) = 5^(1/2) = √5.

Question 7

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

Accepted Answer

Step 1: Rewrite all terms as powers of 2: 2^(x+1) × (2²)^(x−1) = 2³. Step 2: Simplify: 2^(x+1) × 2^(2x−2) = 2³. Step 3: Combine exponents: 2^(x+1+2x−2) = 2³, so 2^(3x−1) = 2³. Step 4: Equate exponents: 3x − 1 = 3, thus 3x = 4, so x = 4/3.

Question 8

Simplify: ∛(27a⁶b⁹) ÷ √(16a⁴b⁴)

Accepted Answer

Step 1: Simplify ∛(27a⁶b⁹) = ∛27 × ∛(a⁶) × ∛(b⁹) = 3 × a² × b³ = 3a²b³. Step 2: Simplify √(16a⁴b⁴) = √16 × √(a⁴) × √(b⁴) = 4 × a² × b² = 4a²b². Step 3: Divide: (3a²b³) ÷ (4a²b²) = (3/4)b.

Question 9

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

Accepted Answer

Step 1: Convert to prime base 2: 16 = 2⁴, 8 = 2³, 4 = 2². Step 2: Rewrite: (2⁴)^(3/4) × (2³)^(2/3) / (2²)^(1/2) = 2³ × 2² / 2¹. Step 3: Combine exponents: 2^(3+2−1) = 2⁴ = 16.

Question 10

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) = ln(k)/ln(3). From 5^y = k, we get y = log₅(k) = ln(k)/ln(5). From 15^z = k, we get z = log₁₅(k) = ln(k)/ln(15). Step 3: Note that 15 = 3 × 5, so ln(15) = ln(3) + ln(5). Step 4: Therefore 1/z = ln(15)/ln(k) = [ln(3) + ln(5)]/ln(k) = ln(3)/ln(k) + ln(5)/ln(k) = 1/x + 1/y. Step 5: Thus 1/z = 1/x + 1/y.

Question 11

If (2^a × 3^b × 5^c)^(1/6) = 2^(1/3) × 3^(1/6) × 5^(1/12), then find the value of 12a + 24b + 36c.

Accepted Answer

Step 1: Expand the left side: 2^(a/6) × 3^(b/6) × 5^(c/6) = 2^(1/3) × 3^(1/6) × 5^(1/12). Step 2: Equate exponents of like bases. For base 2: a/6 = 1/3, so a = 2. For base 3: b/6 = 1/6, so b = 1. For base 5: c/6 = 1/12, so c = 1/2. Step 3: Calculate 12a + 24b + 36c = 12(2) + 24(1) + 36(1/2) = 24 + 24 + 18 = 66.

Question 12

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

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: Now a and b are roots of the quadratic 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. Step 6: Therefore a = 3/2, b = 1/2 (or vice versa), and a + b = 2.

IBPS RRB PO Surds & Indices

IBPS RRB PO Surds & Indices — Past Exam Questions

Surds & Indices— Rules & Concept

Key Points to Remember

Exam-Specific Tips

60-Second Revision — Surds & Indices