Find the cube root of the following number by prime factorization method: 10648

To find the cube root of 10648 using the prime factorization method, we first need to break down the number into its prime factors. This method is effective because it allows us to express the number in terms of its prime components, making it easier to identify perfect cubes.

Step 1: Prime Factorization

We start by dividing 10648 by the smallest prime number, which is 2, and continue dividing until we can no longer do so.

• 10648 ÷ 2 = 5324
• 5324 ÷ 2 = 2662
• 2662 ÷ 2 = 1331

At this point, 1331 is no longer divisible by 2, so we move to the next prime number, which is 3. Since 1331 is not divisible by 3, we try the next prime number, which is 5. Again, 1331 is not divisible by 5. We then check 7 and find that it is also not divisible. Next, we try 11:

• 1331 ÷ 11 = 121

Now we can factor 121, which is a perfect square:

• 121 = 11 × 11

Putting this all together, we have:

• 10648 = 2 × 2 × 2 × 11 × 11 × 11

Step 2: Grouping the Factors

Now that we have the prime factorization, we can express it in a more manageable form:

• 10648 = 2³ × 11³

This representation shows that both 2 and 11 are raised to the power of 3, indicating that they are perfect cubes.

Step 3: Finding the Cube Root

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

• Cube root of 2³ is 2
• Cube root of 11³ is 11

Thus, the cube root of 10648 can be calculated as follows:

• Cube root of 10648 = 2 × 11 = 22

Final Result

The cube root of 10648 is 22. This method not only provides the answer but also reinforces the understanding of how numbers can be broken down into their fundamental components.