Understanding the Cube Root of 3375 Through Factorization

When dealing with mathematical problems, especially those involving cube roots, it's essential to understand the underlying principles and techniques. This article will guide you through the process of finding the cube root of 3375 using factorization. We'll break down the problem into simpler steps and explain each part in detail, ensuring you can follow along and apply the method to similar problems.

Breaking Down 3375

First, we need to understand the prime factorization of 3375. Prime factorization involves breaking down a number into its smallest prime factors. For 3375, the process looks like this:

3375 3 × 1125 3 × 3 × 375 3 × 3 × 3 × 125 3 × 3 × 3 × 5 × 5 × 5

So, the prime factorization of 3375 is 3 × 3 × 3 × 5 × 5 × 5.

Determining the Cube Root Through Factorization

To find the cube root of 3375, we need to find a number that, when cubed, equals 3375. Let's use the prime factorization to solve this problem step by step:

Identify the prime factors: 3375 3 × 3 × 3 × 5 × 5 × 5. Notice that the prime factors can be grouped into sets of three because we are dealing with cube roots. Specifically, we have: 3 × 3 × 3 33 5 × 5 × 5 53 Take the cube root of each set of three factors: Cube root of 33 is 3. Cube root of 53 is 5. Multiply these results together: 3 × 5 15.

Therefore, the cube root of 3375 is 15.

Alternative Methods for Verifying the Answer

While the method above is straightforward, there are other ways to verify the answer. Let's explore a couple of these methods:

Method 1: Using Divisibility Rules

If we use the rule that 3375 is divisible by 3:

Divide 3375 by 3: 1125. Divide 1125 by 3: 375. Divide 375 by 3: 125. Factorize 125: 5 × 5 × 5. Multiply the results: 3 × 3 × 3 × 5 × 5 × 5. Take the cube root: 3 × 5 15.

Method 2: Using Known Cube Numbers

Another method involves using known cube numbers:

Note that 3375 can be expressed as 27000 ÷ 8. Find the cube root of 27000: ?27000 30. Find the cube root of 8: ?8 2. Divide 30 by 2: 30 ÷ 2 15.

Both methods confirm that the cube root of 3375 is indeed 15.

Conclusion

By understanding and applying factorization techniques, you can easily determine the cube root of 3375. This process not only verifies the result but also provides a deeper insight into the properties of numbers and their relationships. Mastery of these techniques can significantly enhance your mathematical problem-solving skills, making it easier to tackle more complex problems in the future.