Find the cube root of 74088.

To find the cube root of 74088, we want to determine which number, when multiplied by itself three times, equals 74088. This process can be approached in a few different ways, including estimation, prime factorization, or using a calculator. Let’s break it down step by step.

Using Prime Factorization

One effective method for finding cube roots is through prime factorization. This involves breaking down the number into its prime factors. Here’s how we can do it:

• Start by dividing 74088 by the smallest prime number, which is 2.
• Continue dividing by 2 until you can no longer do so.
• Then, move to the next smallest prime number, which is 3, and repeat the process.
• Continue this until you have fully factored the number.

Let’s perform the factorization:

• 74088 ÷ 2 = 37044
• 37044 ÷ 2 = 18522
• 18522 ÷ 2 = 9261
• 9261 ÷ 3 = 3087
• 3087 ÷ 3 = 1029
• 1029 ÷ 3 = 343
• 343 ÷ 7 = 49
• 49 ÷ 7 = 7
• 7 ÷ 7 = 1

Now, we can summarize the prime factorization of 74088:

74088 = 2^3 × 3^3 × 7^3

Finding the Cube Root

To find the cube root, we take each prime factor and divide the exponent by 3:

• For 2^3, the cube root is 2.
• For 3^3, the cube root is 3.
• For 7^3, the cube root is 7.

Now, we multiply these results together:

Cube root of 74088 = 2 × 3 × 7

Calculating that gives:

2 × 3 = 6

6 × 7 = 42

Final Result

Thus, the cube root of 74088 is 42. This means that if you multiply 42 by itself three times (42 × 42 × 42), you will indeed get 74088. This method not only provides the answer but also reinforces your understanding of prime factorization and cube roots.