Find the cube root of the following number by prime factorization method: 91125

• a) 45
• b) 35
• c) 55
• d) 465

To find the cube root of 91125 using the prime factorization method, we first need to factor the number into its prime components.

Step 1: Prime Factorization

We start dividing 91125 by the smallest prime numbers:

• 91125 ÷ 3 = 30375
• 30375 ÷ 3 = 10125
• 10125 ÷ 3 = 3375
• 3375 ÷ 3 = 1125
• 1125 ÷ 3 = 375
• 375 ÷ 3 = 125
• 125 ÷ 5 = 25
• 25 ÷ 5 = 5
• 5 ÷ 5 = 1

Resulting Prime Factors

The complete prime factorization of 91125 is:

91125 = 3^6 × 5^3

Step 2: Finding the Cube Root

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

• For 3^6: 6 ÷ 3 = 2, so we take 3^2 = 9
• For 5^3: 3 ÷ 3 = 1, so we take 5^1 = 5

Calculating the Cube Root

Now, we multiply the results:

Cube root of 91125 = 3^2 × 5 = 9 × 5 = 45

Final Answer

The cube root of 91125 is 45.