Prime Factorization of 343

Oct 13, 2025 by thesweetdesigns.com 35 views

Introduction

In mathematics, the cube root of a number is a value that, when multiplied by itself twice, gives the original number. In this article, we will evaluate the cube root of 343, which is denoted as $\sqrt[3]{343}$ . We will explore the different options available and determine the correct answer.

Understanding Cube Roots

A cube root is a type of root that is used to find the value that, when multiplied by itself twice, gives the original number. For example, the cube root of 27 is 3, because 3 multiplied by itself twice equals 27 (3 × 3 × 3 = 27). The cube root of a number is denoted by the symbol $\sqrt[3]{x}$ , where x is the number.

Evaluating $\sqrt[3]{343}$

To evaluate the cube root of 343, we need to find the value that, when multiplied by itself twice, gives 343. We can start by finding the prime factorization of 343.

Prime Factorization of 343

The prime factorization of 343 is 7 × 7 × 7, or $7^3$ . This means that 343 can be expressed as the product of three 7s.

Evaluating the Cube Root

Now that we have the prime factorization of 343, we can evaluate the cube root. Since 343 is equal to $7^3$ , the cube root of 343 is simply 7.

Conclusion

In conclusion, the cube root of 343 is 7. This is because 7 multiplied by itself twice equals 343 (7 × 7 × 7 = 343). Therefore, the correct answer is option B.

Options

A. 171.5
B. 7
C. 114 \frac{1}{3}

Answer

The correct answer is option B. 7.

Additional Tips and Tricks

When evaluating cube roots, it is often helpful to find the prime factorization of the number.
The cube root of a number can be expressed as the product of three identical factors.
To evaluate the cube root of a number, simply find the value that, when multiplied by itself twice, gives the original number.

Real-World Applications

Cube roots have many real-world applications, including:

Geometry: Cube roots are used to find the length of the sides of a cube.
Physics: Cube roots are used to find the volume of a cube.
Engineering: Cube roots are used to find the dimensions of a cube.

Conclusion

Introduction

In our previous article, we evaluated the cube root of 343 and determined that the correct answer is option B. 7. In this article, we will provide a Q&A guide to help you better understand cube roots and how to evaluate them.

Q: What is a cube root?

A: A cube root is a type of root that is used to find the value that, when multiplied by itself twice, gives the original number.

Q: How do I evaluate a cube root?

A: To evaluate a cube root, you need to find the value that, when multiplied by itself twice, gives the original number. You can start by finding the prime factorization of the number.

Q: What is prime factorization?

A: Prime factorization is the process of breaking down a number into its prime factors. For example, the prime factorization of 343 is 7 × 7 × 7, or $7^3$ .

Q: How do I find the prime factorization of a number?

A: To find the prime factorization of a number, you need to break it down into its prime factors. You can do this by dividing the number by prime numbers until you reach 1.

Q: What are some common cube roots?

A: Some common cube roots include:

$\sqrt[3]{8}$ = 2
$\sqrt[3]{27}$ = 3
$\sqrt[3]{64}$ = 4
$\sqrt[3]{125}$ = 5

Q: How do I use cube roots in real-world applications?

A: Cube roots have many real-world applications, including:

Geometry: Cube roots are used to find the length of the sides of a cube.
Physics: Cube roots are used to find the volume of a cube.
Engineering: Cube roots are used to find the dimensions of a cube.

Q: What are some common mistakes to avoid when evaluating cube roots?

A: Some common mistakes to avoid when evaluating cube roots include:

Not finding the prime factorization of the number: Make sure to find the prime factorization of the number before evaluating the cube root.
Not multiplying the factors together: Make sure to multiply the factors together to get the correct answer.
Not checking the answer: Make sure to check the answer to make sure it is correct.

Q: How can I practice evaluating cube roots?

A: You can practice evaluating cube roots by:

Solving problems: Try solving problems that involve evaluating cube roots.
Using online resources: Use online resources such as calculators or worksheets to practice evaluating cube roots.
Working with a tutor: Work with a tutor to practice evaluating cube roots.

Conclusion

In conclusion, evaluating cube roots is an important concept in mathematics. By understanding how to evaluate cube roots, you can solve a wide range of problems in geometry, physics, and engineering. In this article, we provided a Q&A guide to help you better understand cube roots and how to evaluate them.