20736 power 1/8

20736^(1)/(8) Step 1: Find the prime factorization of 20736. 20736 = 2^10 × 3^4 Step 2: Rewrite the expression using the prime factorization. (2^10 × 3^4)^(1)/(8) Step 3: Apply the exponent rule (a^m b^n)^p = a^mp b^np. 2^10 × (1)/(8) × 3^4 × (1)/(8) Step 4: Simplify the exponents. 2^(10)/(8) × 3^(4)/(8) 2^(5)/(4) × 3^(1)/(2) Step 5: Rewrite the expression using radicals. 2^(5)/(4) = 2^1 + (1)/(4) = 2 × 2^(1)/(4) = 2[4]2 3^(1)/(2) = sqrt(3) Step 6: Combine the terms. 2[4]2 × sqrt(3) Alternatively, we can look for a number that, when raised to the power of 8, equals 20736. Let x = 20736^(1)/(8). Then x^8 = 20736. We can test small integer values. 1^8 = 1 2^8 = 256 3^8 = 6561 4^8 = 65536 Since 3^8 < 20736 < 4^8, the answer is not an integer. Let's re-examine the prime factorization: 20736 = 2^10 × 3^4 20736^(1)/(8) = (2^10 × 3^4)^(1)/(8) = 2^(10)/(8) × 3^(4)/(8) = 2^(5)/(4) × 3^(1)/(2) 2^(5)/(4) = 2^1.25 3^(1)/(2) = sqrt(3) ≈ 1.732 2^1.25 = 2 × 2^0.25 = 2 × [4]2 [4]2 ≈ 1.189 2 × 1.189 ≈ 2.378 2.378 × 1.732 ≈ 4.119 Let's check if there was a simpler way to factor 20736. 20736 = 144^2 = (12^2)^2 = 12^4 = (2^2 × 3)^4 = 2^8 × 3^4. Ah, there was a mistake in the initial prime factorization. Let's correct the prime factorization: 20736 = 144 × 144 144 = 12 × 12 = (2^2 × 3) × (2^2 × 3) = 2^4 × 3^2 So, 20736 = (2^4 × 3^2) × (2^4 × 3^2) = 2^8 × 3^4. Now, let's recalculate: 20736^(1)/(8) = (2^8 × 3^4)^(1)/(8) Step 3 (corrected): Apply the exponent rule (a^m b^n)^p = a^mp b^np. 2^8 × (1)/(8) × 3^4 × (1)/(8) Step 4 (corrected): Simplify the exponents. 2^1 × 3^(4)/(8) 2 × 3^(1)/(2) Step 5 (corrected): Rewrite the expression using radicals. 2 × sqrt(3) The final answer is 2sqrt(3).