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

To find the cube root of 512 using the prime factorization method, we first need to break down the number into its prime factors. This method is quite effective because it allows us to see the building blocks of the number clearly, making it easier to extract roots.

Step 1: Prime Factorization of 512

Let's start by dividing 512 by the smallest prime number, which is 2. We continue dividing by 2 until we can no longer do so:

• 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

From this process, we can see that 512 can be expressed as:

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

Or, more compactly, we can write it as:

512 = 2^9

Step 2: Finding the Cube Root

The cube root of a number is the value that, when multiplied by itself three times, gives the original number. In terms of prime factorization, we can find the cube root by taking each prime factor and dividing its exponent by 3.

For our case:

Cube root of 512 = Cube root of (2^9)

Now, we divide the exponent by 3:

9 ÷ 3 = 3

This tells us that:

Cube root of 512 = 2^3

Step 3: Calculating the Final Value

Now, we simply calculate \(2^3\):

2^3 = 2 × 2 × 2 = 8

Final Result

Thus, the cube root of 512 is 8.

This method not only provides the answer but also reinforces your understanding of how numbers can be broken down into their fundamental components. Prime factorization is a powerful tool in mathematics, especially when dealing with roots and exponents.