Question 1

Simplify: (3⁴ × 3²) ÷ 3³

Accepted Answer

Step 1: Use the law of indices: a^m × a^n = a^(m+n). Step 2: 3⁴ × 3² = 3^(4+2) = 3⁶. Step 3: Use a^m ÷ a^n = a^(m-n). Step 4: 3⁶ ÷ 3³ = 3^(6-3) = 3³ = 27. Therefore, the answer is 27.

Question 2

Rationalize the denominator: 1/(√7 - √2)

Accepted Answer

Step 1: Multiply numerator and denominator by the conjugate (√7 + √2). Step 2: Numerator becomes: 1 × (√7 + √2) = √7 + √2. Step 3: Denominator becomes: (√7 - √2)(√7 + √2) = 7 - 2 = 5 (using difference of squares). Step 4: Result = (√7 + √2)/5. Therefore, the answer is (√7 + √2)/5.

Question 3

Simplify: ∛(27 × 8)

Accepted Answer

Step 1: Use the property ∛(a × b) = ∛a × ∛b. Step 2: ∛(27 × 8) = ∛27 × ∛8. Step 3: ∛27 = 3 (since 3³ = 27) and ∛8 = 2 (since 2³ = 8). Step 4: 3 × 2 = 6. Therefore, the answer is 6.

Question 4

Evaluate: (5⁰ + 6⁰ + 7⁰) × 10

Accepted Answer

Step 1: Recall that any non-zero number raised to the power 0 equals 1. Step 2: 5⁰ = 1, 6⁰ = 1, 7⁰ = 1. Step 3: 5⁰ + 6⁰ + 7⁰ = 1 + 1 + 1 = 3. Step 4: 3 × 10 = 30. Therefore, the answer is 30.

Question 5

Simplify: (√5 + √3)(√5 - √3)

Accepted Answer

Step 1: Use the identity (a + b)(a - b) = a² - b². Step 2: Here a = √5 and b = √3. Step 3: (√5)² - (√3)² = 5 - 3 = 2. Therefore, the answer is 2.

Question 6

If 2^(3x) = 64, find the value of 2^(2x - 1).

Accepted Answer

Step 1: Solve for x from 2^(3x) = 64. Since 64 = 2^6, we have 3x = 6, so x = 2. Step 2: Substitute x = 2 into 2^(2x - 1) = 2^(2(2) - 1) = 2^(4 - 1) = 2^3 = 8.

Question 7

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

Accepted Answer

Step 1: Use the product rule in the numerator: 2^(2/3) × 2^(4/3) = 2^(2/3 + 4/3) = 2^(6/3) = 2^2 = 4. Step 2: Divide by 2^(1/3): 4 / 2^(1/3) = 2^2 / 2^(1/3). Step 3: Apply the quotient rule: 2^(2 - 1/3) = 2^(6/3 - 1/3) = 2^(5/3). Step 4: Express as a radical: 2^(5/3) = ∛(2^5) = ∛32.

Question 8

If √(2x + 3) = 5, find the value of x.

Accepted Answer

Step 1: Square both sides of the equation: (√(2x + 3))² = 5². Step 2: This gives 2x + 3 = 25. Step 3: Solve for x: 2x = 25 - 3 = 22, so x = 11. Step 4: Verify: √(2(11) + 3) = √(22 + 3) = √25 = 5 ✓

Question 9

Rationalize: 1 / (√5 - √3). Express in the form (a√5 + b√3) / c where a, b, c are integers.

Accepted Answer

Step 1: Multiply numerator and denominator by the conjugate (√5 + √3): [1 × (√5 + √3)] / [(√5 - √3)(√5 + √3)]. Step 2: Denominator = (√5)² - (√3)² = 5 - 3 = 2. Step 3: Result = (√5 + √3) / 2. Therefore a = 1, b = 1, c = 2, and a + b + c = 4.

Question 10

Simplify: (√8 + √18) / √2

Accepted Answer

Step 1: Simplify √8 = √(4×2) = 2√2 and √18 = √(9×2) = 3√2. Step 2: Substitute into the expression: (2√2 + 3√2) / √2 = 5√2 / √2. Step 3: Divide: 5√2 / √2 = 5.

Question 11

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

Accepted Answer

Step 1: Simplify the numerator exponent: 5^(2/3) × 5^(1/3) = 5^(2/3 + 1/3) = 5^1 = 5. Step 2: Raise to power 3: (5)^3 = 125. Step 3: Simplify the denominator: (5^2)^(1/2) = 5^(2 × 1/2) = 5^1 = 5. Step 4: Divide: 125 ÷ 5 = 25.

Question 12

If √(5 + √24) = √a + √b, where a and b are positive integers, find a + b.

Accepted Answer

Step 1: Assume √(5 + √24) = √a + √b. Step 2: Square both sides: 5 + √24 = a + b + 2√(ab). Step 3: Compare rational and irrational parts: a + b = 5 and 2√(ab) = √24. Step 4: From 2√(ab) = √24, square: 4ab = 24, so ab = 6. Step 5: Solve a + b = 5 and ab = 6: a and b are roots of x² - 5x + 6 = 0, giving (x-2)(x-3) = 0, so a = 3, b = 2. Step 6: a + b = 5.

Question 13

Rationalize and simplify: (3 + √5) / (3 - √5) + (3 - √5) / (3 + √5)

Accepted Answer

Step 1: Rationalize the first fraction: (3 + √5) / (3 - √5) × (3 + √5) / (3 + √5) = (3 + √5)² / (9 - 5) = (9 + 6√5 + 5) / 4 = (14 + 6√5) / 4. Step 2: Rationalize the second fraction: (3 - √5) / (3 + √5) × (3 - √5) / (3 - √5) = (3 - √5)² / (9 - 5) = (9 - 6√5 + 5) / 4 = (14 - 6√5) / 4. Step 3: Add: (14 + 6√5) / 4 + (14 - 6√5) / 4 = 28 / 4 = 7.

SSC GD Constable Surds & Indices

Surds & Indices— Rules & Concept

Key Points to Remember

Exam-Specific Tips

Surds & Indices — Practice Questions

60-Second Revision — Surds & Indices