Find the cube root of 5832.

To find the cube root of 5832, we can approach the problem step by step. The cube root of a number is a value that, when multiplied by itself three times, gives the original number. In this case, we are looking for a number \( x \) such that \( x^3 = 5832 \).

Breaking Down the Problem

One effective way to find the cube root is to factor the number into its prime components. This method allows us to see the underlying structure of the number, making it easier to determine the cube root.

Prime Factorization of 5832

Let’s start by dividing 5832 by the smallest prime number, which is 2:

• 5832 ÷ 2 = 2916
• 2916 ÷ 2 = 1458
• 1458 ÷ 2 = 729

Now, 729 is not divisible by 2, so we move to the next prime number, which is 3:

• 729 ÷ 3 = 243
• 243 ÷ 3 = 81
• 81 ÷ 3 = 27
• 27 ÷ 3 = 9
• 9 ÷ 3 = 3
• 3 ÷ 3 = 1

Putting It All Together

Now, we can summarize the prime factorization of 5832:

5832 = 2^3 × 3^6

Finding the Cube Root

To find the cube root, we take the cube root of each prime factor:

• The cube root of \( 2^3 \) is \( 2 \).
• The cube root of \( 3^6 \) is \( 3^{6/3} = 3^2 = 9 \).

Now, we multiply these results together:

Cube root of 5832 = \( 2 \times 9 = 18 \).

Final Result

Thus, the cube root of 5832 is 18. To verify, you can check that \( 18^3 = 5832 \), confirming that our calculations are correct.