Step 1: Understand the notation: a^(m/n) = (ⁿ√a)^m or ⁿ√(a^m)

Step 2: Choose the easier approach: 8^(2/3) = (∛8)²

Step 3: Find the cube root: ∛8 = 2

Step 4: Square the result: 2² = 4

Answer: 4

Method 1: Take the root first, then the power $16^{\frac{3}{4}} = (\sqrt[4]{16})^3 = 2^3 = 8$

Step 1: Use product rule: x^m · x^n = x^(m+n)

Step 2: Add the exponents: x^(3/4) · x^(1/4) = x^(3/4 + 1/4)

Step 3: Add fractions: 3/4 + 1/4 = 4/4 = 1

Step 4: Simplify: x^1 = x

Answer: x

Step 1: Raise both sides to the reciprocal power: The reciprocal of 3/2 is 2/3 (x^(3/2))^(2/3) = 27^(2/3)

Step 2: Simplify left side using power rule: x^((3/2)·(2/3)) = 27^(2/3) x^1 = 27^(2/3) x = 27^(2/3)

Step 3: Evaluate 27^(2/3): 27^(2/3) = (∛27)² = 3² = 9

Step 4: Check: 9^(3/2) = (√9)³ = 3³ = 27 ✓

Answer: x = 9

Use the quotient rule: $\frac{a^m}{a^n} = a^{m-n}$

Step 1: Apply power to each factor: (16x^8y^4)^(3/4) = 16^(3/4) · (x^8)^(3/4) · (y^4)^(3/4)

Step 2: Evaluate 16^(3/4): 16^(3/4) = (⁴√16)³ = 2³ = 8

Step 3: Apply power rule to x: (x^8)^(3/4) = x^(8·3/4) = x^(24/4) = x^6

Step 4: Apply power rule to y: (y^4)^(3/4) = y^(4·3/4) = y^(12/4) = y^3

Step 5: Combine: 8x^6y^3

Answer: 8x^6y^3