Simplify: 216 343 3 \sqrt[3]{\frac{216}{343}} 3 343 216 ​ ​ A. 1 2 \frac{1}{2} 2 1 ​ B. 6 7 \frac{6}{7} 7 6 ​ C. 216 343 \frac{216}{343} 343 216 ​

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Understanding the Problem

To simplify the given expression, we need to evaluate the cube root of the fraction $\frac{216}{343}$. This involves breaking down the numerator and denominator into their prime factors and then simplifying the resulting expression.

Breaking Down the Numerator and Denominator

The numerator, 216, can be broken down into its prime factors as follows:

$216 = 2^3 \times 3^3$

The denominator, 343, can be broken down into its prime factors as follows:

$343 = 7^3$

Simplifying the Expression

Now that we have the prime factors of the numerator and denominator, we can simplify the expression by canceling out any common factors.

$\sqrt[3]{\frac{216}{343}} = \sqrt[3]{\frac{2^3 \times 3^3}{7^3}}$

Canceling Out Common Factors

Since the numerator and denominator both have a cube of 3, we can cancel out the cube of 3 in the numerator and denominator.

$\sqrt[3]{\frac{2^3 \times 3^3}{7^3}} = \sqrt[3]{\frac{2^3}{7^3}}$

Evaluating the Cube Root

Now that we have simplified the expression, we can evaluate the cube root of the fraction.

$\sqrt[3]{\frac{2^3}{7^3}} = \frac{2}{7}$

Conclusion

Therefore, the simplified expression is $\frac{2}{7}$.

Comparison with Answer Choices

Let's compare our simplified expression with the answer choices provided.

A. $\frac{1}{2}$

B. $\frac{6}{7}$

C. $\frac{216}{343}$

Our simplified expression, $\frac{2}{7}$, does not match any of the answer choices. However, we can see that answer choice B, $\frac{6}{7}$, is close to our simplified expression. But, it is not the correct answer.

Final Answer

The final answer is $\boxed{\frac{2}{7}}$.

Step-by-Step Solution

Here's a step-by-step solution to the problem:

1. Break down the numerator and denominator into their prime factors.
2. Simplify the expression by canceling out any common factors.
3. Evaluate the cube root of the fraction.
4. Compare the simplified expression with the answer choices.

Tips and Tricks

Here are some tips and tricks to help you solve this problem:

• Make sure to break down the numerator and denominator into their prime factors.
• Simplify the expression by canceling out any common factors.
• Evaluate the cube root of the fraction carefully.
• Compare the simplified expression with the answer choices carefully.

Common Mistakes

Here are some common mistakes to avoid when solving this problem:

• Failing to break down the numerator and denominator into their prime factors.
• Failing to simplify the expression by canceling out any common factors.
• Evaluating the cube root of the fraction incorrectly.
• Comparing the simplified expression with the answer choices incorrectly.

Real-World Applications

This problem has real-world applications in various fields such as engineering, physics, and mathematics. For example, in engineering, you may need to calculate the cube root of a fraction to determine the volume of a cube. In physics, you may need to calculate the cube root of a fraction to determine the velocity of an object. In mathematics, you may need to calculate the cube root of a fraction to determine the value of a mathematical expression.

Conclusion

In conclusion, simplifying the expression $\sqrt[3]{\frac{216}{343}}$ involves breaking down the numerator and denominator into their prime factors, simplifying the expression by canceling out any common factors, evaluating the cube root of the fraction, and comparing the simplified expression with the answer choices. The final answer is $\boxed{\frac{2}{7}}$.

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Frequently Asked Questions

Q: What is the cube root of a fraction?

A: The cube root of a fraction is a mathematical operation that involves finding the cube root of the numerator and the cube root of the denominator separately and then simplifying the resulting expression.

Q: How do I simplify the cube root of a fraction?

A: To simplify the cube root of a fraction, you need to break down the numerator and denominator into their prime factors, simplify the expression by canceling out any common factors, and then evaluate the cube root of the fraction.

Q: What are the prime factors of 216 and 343?

A: The prime factors of 216 are $2^3 \times 3^3$, and the prime factors of 343 are $7^3$.

Q: How do I simplify the expression $\sqrt[3]{\frac{216}{343}}$?

A: To simplify the expression $\sqrt[3]{\frac{216}{343}}$, you need to break down the numerator and denominator into their prime factors, simplify the expression by canceling out any common factors, and then evaluate the cube root of the fraction.

Q: What is the final answer to the expression $\sqrt[3]{\frac{216}{343}}$?

A: The final answer to the expression $\sqrt[3]{\frac{216}{343}}$ is $\boxed{\frac{2}{7}}$.

Q: What are some common mistakes to avoid when simplifying the cube root of a fraction?

A: Some common mistakes to avoid when simplifying the cube root of a fraction include failing to break down the numerator and denominator into their prime factors, failing to simplify the expression by canceling out any common factors, and evaluating the cube root of the fraction incorrectly.

Q: What are some real-world applications of simplifying the cube root of a fraction?

A: Some real-world applications of simplifying the cube root of a fraction include calculating the volume of a cube, determining the velocity of an object, and solving mathematical expressions.

Q: How do I evaluate the cube root of a fraction?

A: To evaluate the cube root of a fraction, you need to find the cube root of the numerator and the cube root of the denominator separately and then simplify the resulting expression.

Q: What is the difference between the cube root of a fraction and the cube root of a number?

A: The cube root of a fraction is a mathematical operation that involves finding the cube root of the numerator and the cube root of the denominator separately and then simplifying the resulting expression. The cube root of a number is a mathematical operation that involves finding the cube root of a single number.

Q: How do I compare the simplified expression with the answer choices?

A: To compare the simplified expression with the answer choices, you need to evaluate the simplified expression and then compare it with the answer choices.

Q: What are some tips and tricks for simplifying the cube root of a fraction?

A: Some tips and tricks for simplifying the cube root of a fraction include breaking down the numerator and denominator into their prime factors, simplifying the expression by canceling out any common factors, and evaluating the cube root of the fraction carefully.

Conclusion

In conclusion, simplifying the expression $\sqrt[3]{\frac{216}{343}}$ involves breaking down the numerator and denominator into their prime factors, simplifying the expression by canceling out any common factors, evaluating the cube root of the fraction, and comparing the simplified expression with the answer choices. The final answer is $\boxed{\frac{2}{7}}$.