Find the cube root of 5832 by the prime factorization method.

To find the cube root of 5832 using the prime factorization method, we first need to break down the number into its prime factors.

Step 1: Prime Factorization

We start 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

Step 2: Listing the Prime Factors

The complete prime factorization of 5832 is:

• 2 × 2 × 2 × 3 × 3 × 3 × 3 × 3 = 2³ × 3⁶

Step 3: Finding the Cube Root

To find the cube root, we take one-third of the powers of the prime factors:

• For 2³, the cube root is 2^(3/3) = 2¹ = 2
• For 3⁶, the cube root is 3^(6/3) = 3² = 9

Final Calculation

Now, we multiply the results of the cube roots:

Cube root of 5832 = 2 × 9 = 18

Thus, the cube root of 5832 is 18.