Question 1

If √(x + 5) = 7, find the value of 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 2

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

Accepted Answer

Step 1: Express 512 as a power of 2: 512 = 2^9. Step 2: Set up equation: 2^(3x) = 2^9. Step 3: Equate exponents: 3x = 9. Step 4: Solve: x = 3.

Question 3

Simplify: ∛(27a⁶b³) / ∛(8a³b⁶)

Accepted Answer

Step 1: Rewrite as ∛(27a⁶b³) ÷ ∛(8a³b⁶) = ∛[(27a⁶b³)/(8a³b⁶)]. Step 2: Simplify inside: (27/8) × (a⁶/a³) × (b³/b⁶) = (27/8) × a³ × b⁻³. Step 3: Take cube root: ∛(27/8) × ∛(a³) × ∛(b⁻³) = (3/2) × a × (1/b) = 3a/(2b).

Question 4

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

Accepted Answer

Step 1: Convert to base 2: 16 = 2⁴, so 16^(1/4) = (2⁴)^(1/4) = 2¹ = 2. Step 2: 8 = 2³, so 8^(1/3) = (2³)^(1/3) = 2¹ = 2. Step 3: Numerator = 2 × 2 = 4 = 2². Step 4: Divide: 2² / 2^(1/2) = 2^(2 - 1/2) = 2^(3/2) = 2√2.

Question 5

Rationalize: 1/(2 + √3) and find the denominator after rationalization.

Accepted Answer

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

Question 6

Simplify: (√5 + √2)² - (√5 - √2)²

Accepted Answer

Step 1: Use the identity (a + b)² - (a - b)² = 4ab. Step 2: Here, a = √5 and b = √2. Step 3: 4 × √5 × √2 = 4√10.

Question 7

If 3^(2x) = 729, find x.

Accepted Answer

Step 1: Express 729 as a power of 3. 729 = 3⁶ (since 3 × 3 × 3 × 3 × 3 × 3 = 729). Step 2: Rewrite: 3^(2x) = 3⁶. Step 3: Equate exponents: 2x = 6. Step 4: x = 3.

Question 8

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

Accepted Answer

Step 1: Convert each term to prime base powers. 16^(1/4) = (2⁴)^(1/4) = 2. 27^(1/3) = (3³)^(1/3) = 3. 32^(1/5) = (2⁵)^(1/5) = 2. Step 2: Substitute: (2 × 3) / 2 = 6 / 2 = 3.

Question 9

Simplify: (∛64 × ⁴√81) ÷ ⁵√32

Accepted Answer

Step 1: Express each term as powers. ∛64 = ∛(4³) = 4. ⁴√81 = ⁴√(3⁴) = 3. ⁵√32 = ⁵√(2⁵) = 2. Step 2: Substitute: (4 × 3) ÷ 2 = 12 ÷ 2 = 6.

Question 10

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

Accepted Answer

Step 1: Express all terms as powers of 2. 4^(x-1) = (2²)^(x-1) = 2^(2x-2). 64 = 2⁶. Step 2: Rewrite: 2^(x+1) × 2^(2x-2) = 2⁶. Step 3: Using a^m × a^n = a^(m+n): 2^(x+1+2x-2) = 2⁶, so 2^(3x-1) = 2⁶. Step 4: Equate exponents: 3x - 1 = 6, so 3x = 7, thus x = 7/3.

Question 11

Simplify: (5^(1/3) × 5^(1/6)) / 5^(1/2)

Accepted Answer

Step 1: Use the rule a^m × a^n = a^(m+n). Numerator: 5^(1/3 + 1/6) = 5^(2/6 + 1/6) = 5^(3/6) = 5^(1/2). Step 2: Divide: 5^(1/2) / 5^(1/2) = 5^(1/2 - 1/2) = 5^0 = 1.

Question 12

If (∜x)^8 = 256, find x.

Accepted Answer

Step 1: (∜x)^8 = (x^(1/4))^8 = x^(8/4) = x^2. Step 2: x^2 = 256. Step 3: x = ±16. Since we typically take the positive root in such problems, x = 16.

Question 13

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

Accepted Answer

Step 1: Numerator: 3^(2/3 + 1/3) = 3^(3/3) = 3^1 = 3. Step 2: Denominator: 3^(1/6 + 1/6) = 3^(2/6) = 3^(1/3). Step 3: Divide: 3 / 3^(1/3) = 3^1 / 3^(1/3) = 3^(1 - 1/3) = 3^(2/3).

Question 14

If 3^a = 27 and 2^b = 32, find the value of a^b.

Accepted Answer

Step 1: Solve for a: 3^a = 27 = 3^3, so a = 3. Step 2: Solve for b: 2^b = 32 = 2^5, so b = 5. Step 3: Calculate a^b = 3^5 = 243.

SSC MTS Surds & Indices

Surds & Indices— Rules & Concept

Key Points to Remember

Exam-Specific Tips

Surds & Indices — Practice Questions

60-Second Revision — Surds & Indices