Question 1

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 2

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 3

Simplify: (2⁵ × 3⁴ × 5²) / (2³ × 3² × 5) + √(144 + 81) - 12

Accepted Answer

Step 1: Simplify the fraction using exponent rules.
(2⁵ × 3⁴ × 5²) / (2³ × 3² × 5) = 2^(5-3) × 3^(4-2) × 5^(2-1) = 2² × 3² × 5 = 4 × 9 × 5 = 180

Step 2: Simplify the square root.
√(144 + 81) = √225 = 15

Step 3: Combine all parts.
180 + 15 - 12 = 183

Therefore, the answer is 183.

Question 4

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 5

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 6

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 7

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 8

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 9

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 10

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.

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