Find the cube root of 512 by using the prime factorization method.

To find the cube root of 512 using the prime factorization method, we start by expressing 512 as a product of its prime factors. Let’s break it down step by step:

Step 1: Prime Factorization of 512

We divide 512 repeatedly by its smallest prime number, which is 2:

• 512 ÷ 2 = 256
• 256 ÷ 2 = 128
• 128 ÷ 2 = 64
• 64 ÷ 2 = 32
• 32 ÷ 2 = 16
• 16 ÷ 2 = 8
• 8 ÷ 2 = 4
• 4 ÷ 2 = 2
• 2 ÷ 2 = 1

Thus, the prime factorization of 512 is:

512 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2

Step 2: Grouping Factors into Triplets

To find the cube root of 512, we group the factors into sets of three:

512 = (2 × 2 × 2) × (2 × 2 × 2) × (2 × 2 × 2)

Step 3: Taking the Cube Root

Now, take the cube root of both sides:

∛512 = ∛[(2 × 2 × 2) × (2 × 2 × 2) × (2 × 2 × 2)]

Since the cube root of a product of cubes is the product of their cube roots, we have:

∛512 = 2 × 2 × 2 = 8

Final Answer

Therefore, the cube root of 512 is 8.

Summary

By using the prime factorization method, we grouped the factors of 512 into triplets and calculated the cube root. This step-by-step approach ensures clarity and accuracy. Hence, the final answer to "find the cube root of 512" is 8.