Question 1

Simplify: (∛8 × ∛27) ÷ ∛64

Accepted Answer

Step 1: Calculate ∛8 = 2 (since 2³ = 8). Step 2: Calculate ∛27 = 3 (since 3³ = 27). Step 3: Multiply: 2 × 3 = 6. Step 4: Calculate ∛64 = 4 (since 4³ = 64). Step 5: Divide: 6 ÷ 4 = 1.5 or 3/2.

Question 2

If 2ˣ × 2³ = 2⁸, find x.

Accepted Answer

Step 1: Apply the multiplication rule aᵐ × aⁿ = aᵐ⁺ⁿ: 2ˣ × 2³ = 2ˣ⁺³. Step 2: Set equal to 2⁸: 2ˣ⁺³ = 2⁸. Step 3: Equate exponents: x + 3 = 8. Step 4: Solve for x: x = 8 − 3 = 5.

Question 3

Simplify: (2³)² ÷ 2⁴ × 2⁵

Accepted Answer

Step 1: Apply power rule (aᵐ)ⁿ = aᵐⁿ: (2³)² = 2⁶. Step 2: Rewrite as 2⁶ ÷ 2⁴ × 2⁵. Step 3: Apply division rule aᵐ ÷ aⁿ = aᵐ⁻ⁿ: 2⁶⁻⁴ = 2². Step 4: Apply multiplication rule aᵐ × aⁿ = aᵐ⁺ⁿ: 2² × 2⁵ = 2⁷ = 128.

Question 4

If 3ˣ = 81, find the value of 3ˣ⁻².

Accepted Answer

Step 1: Express 81 as a power of 3: 81 = 3⁴. Step 2: Therefore, 3ˣ = 3⁴, so x = 4. Step 3: Calculate 3ˣ⁻² = 3⁴⁻² = 3² = 9.

Question 5

Simplify: √(16 × 25) ÷ √4

Accepted Answer

Step 1: Use property √(a × b) = √a × √b: √(16 × 25) = √16 × √25 = 4 × 5 = 20. Step 2: Calculate √4 = 2. Step 3: Divide: 20 ÷ 2 = 10.

Question 6

If 5²ˣ = 625, find the value of 5ˣ.

Accepted Answer

Step 1: Express 625 as a power of 5: 625 = 5⁴. Step 2: Therefore, 5²ˣ = 5⁴, so 2x = 4, which gives x = 2. Step 3: Calculate 5ˣ = 5² = 25.

Question 7

Simplify: (16^(1/4) × 27^(1/3)) / (32^(1/5))

Accepted Answer

Step 1: Convert to prime bases: 16 = 2⁴, 27 = 3³, 32 = 2⁵. Step 2: Apply exponents: 16^(1/4) = (2⁴)^(1/4) = 2¹ = 2. Step 3: 27^(1/3) = (3³)^(1/3) = 3¹ = 3. Step 4: 32^(1/5) = (2⁵)^(1/5) = 2¹ = 2. Step 5: Substitute: (2 × 3) / 2 = 6/2 = 3.

Question 8

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

Accepted Answer

Step 1: Square both sides: 2 + √3 = 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: 4ab = 3, so ab = 3/4. Step 4: Solve system: a + b = 2 and ab = 3/4. These are roots of t² − 2t + 3/4 = 0. Step 5: Using quadratic formula: t = (2 ± √(4 − 3))/2 = (2 ± 1)/2. So t = 3/2 or t = 1/2. Step 6: Thus a = 3/2, b = 1/2 (or vice versa). Step 7: a + b = 3/2 + 1/2 = 2.

Question 9

If √(2 + √3) = (√a + √b) / √2, where a and b are positive integers, find a + b.

Accepted Answer

Step 1: Square both sides: 2 + √3 = (√a + √b)^2 / 2. Step 2: Expand right side: 2 + √3 = (a + b + 2√(ab)) / 2. Step 3: Multiply both sides by 2: 4 + 2√3 = a + b + 2√(ab). Step 4: Equate rational and irrational parts: a + b = 4 and 2√(ab) = 2√3. Step 5: From 2√(ab) = 2√3, we get √(ab) = √3, so ab = 3. Step 6: Solve the system: a + b = 4 and ab = 3. These are roots of t^2 - 4t + 3 = 0, which factors as (t - 1)(t - 3) = 0. So a = 3, b = 1 (or vice versa). Step 7: a + b = 4.

Question 10

If (2^a × 3^b × 5^c)^(1/3) = 2^2 × 3^(1/2) × 5^(1/3), then find the value of 6a + 4b + 3c.

Accepted Answer

Step 1: Equate the exponents on both sides after applying the power rule. Left side: 2^(a/3) × 3^(b/3) × 5^(c/3). Right side: 2^2 × 3^(1/2) × 5^(1/3). Step 2: Compare exponents: a/3 = 2 → a = 6; b/3 = 1/2 → b = 3/2; c/3 = 1/3 → c = 1. Step 3: Calculate 6a + 4b + 3c = 6(6) + 4(3/2) + 3(1) = 36 + 6 + 3 = 45.

SBI PO Surds & Indices

Surds & Indices— Rules & Concept

Key Points to Remember

Exam-Specific Tips

Surds & Indices — Practice Questions

60-Second Revision — Surds & Indices