Find the cube root of 46656 by prime factorization method.

To find the cube root of 46656 using the prime factorization method, we first need to break down the number into its prime factors.

Step 1: Prime Factorization

We start dividing 46656 by the smallest prime number, which is 2:

• 46656 ÷ 2 = 23328
• 23328 ÷ 2 = 11664
• 11664 ÷ 2 = 5832
• 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: Collecting the Prime Factors

From the divisions, we can summarize the prime factorization:

46656 = 2^6 × 3^6

Step 3: Finding the Cube Root

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

• Cube root of 2^6 = 2^(6/3) = 2^2 = 4
• Cube root of 3^6 = 3^(6/3) = 3^2 = 9

Final Calculation

Now, we multiply the results:

Cube root of 46656 = 4 × 9 = 36

Result

The cube root of 46656 is 36.