Find the cube root of 2744.

To find the cube root of 2744, we are essentially looking for a number that, when multiplied by itself three times, equals 2744. This process can be approached through various methods, including prime factorization, estimation, or using a calculator. Let's break it down step by step.

Understanding Cube Roots

The cube root of a number \( x \) is a value \( y \) such that \( y^3 = x \). In this case, we want to find \( y \) such that \( y^3 = 2744 \).

Using Prime Factorization

One effective way to find the cube root is through prime factorization. This involves breaking down the number into its prime factors.

• First, we can divide 2744 by the smallest prime number, which is 2:

2744 ÷ 2 = 1372

1372 ÷ 2 = 686

686 ÷ 2 = 343

Now, 343 is not divisible by 2, so we move to the next prime number, which is 7:

343 ÷ 7 = 49

49 ÷ 7 = 7

7 ÷ 7 = 1

Putting It All Together

Now we can summarize the prime factorization of 2744:

2744 = 2^3 × 7^3

Since both factors are raised to the power of 3, we can take the cube root of each factor:

• The cube root of \( 2^3 \) is 2.
• The cube root of \( 7^3 \) is 7.

Therefore, the cube root of 2744 is:

2 × 7 = 14

Verification

To ensure our answer is correct, we can verify by cubing 14:

14 × 14 × 14 = 196 × 14 = 2744

Conclusion

The cube root of 2744 is indeed 14. This method not only provides the answer but also reinforces the concept of prime factorization and how it can be applied to find roots of numbers. If you have any further questions about cube roots or related concepts, feel free to ask!