Evaluate 125 1000 3 \sqrt[3]{\frac{125}{1000}} 3 1000 125 ​ ​ .A. 25 200 \frac{25}{200} 200 25 ​ B. 15 100 \frac{15}{100} 100 15 ​ C. 3 25 \frac{3}{25} 25 3 ​ D. 5 10 \frac{5}{10} 10 5 ​

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Understanding the Problem

To evaluate the given expression, we need to start by simplifying the fraction inside the cube root. The expression $\sqrt[3]{\frac{125}{1000}}$ can be broken down into two parts: the numerator and the denominator. The numerator is 125, and the denominator is 1000.

Simplifying the Fraction

We can simplify the fraction by finding the greatest common divisor (GCD) of the numerator and the denominator. The GCD of 125 and 1000 is 125. We can divide both the numerator and the denominator by 125 to simplify the fraction.

$\frac{125}{1000} = \frac{125 \div 125}{1000 \div 125} = \frac{1}{8}$

Evaluating the Cube Root

Now that we have simplified the fraction, we can evaluate the cube root. The cube root of a fraction can be found by taking the cube root of the numerator and the cube root of the denominator.

$\sqrt[3]{\frac{1}{8}} = \frac{\sqrt[3]{1}}{\sqrt[3]{8}}$

Simplifying the Cube Root

We can simplify the cube root of 1, which is 1. The cube root of 8 can be simplified by finding the cube root of 2, which is 2.

$\frac{\sqrt[3]{1}}{\sqrt[3]{8}} = \frac{1}{2}$

Converting the Fraction to a Decimal

To convert the fraction to a decimal, we can divide the numerator by the denominator.

$\frac{1}{2} = 0.5$

Comparing the Answer Choices

Now that we have evaluated the expression, we can compare our answer to the answer choices.

A. $\frac{25}{200}$

B. $\frac{15}{100}$

C. $\frac{3}{25}$

D. $\frac{5}{10}$

Our answer is $\frac{1}{2}$, which is equivalent to 0.5. We can convert each of the answer choices to a decimal to compare them to our answer.

A. $\frac{25}{200} = 0.125$

B. $\frac{15}{100} = 0.15$

C. $\frac{3}{25} = 0.12$

D. $\frac{5}{10} = 0.5$

Conclusion

Based on our evaluation, the correct answer is D. $\frac{5}{10}$, which is equivalent to 0.5.

Final Answer

The final answer is $\boxed{\frac{5}{10}}$.

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Frequently Asked Questions

Q: What is the greatest common divisor (GCD) of 125 and 1000?

A: The GCD of 125 and 1000 is 125.

Q: How do I simplify the fraction $\frac{125}{1000}$?

A: To simplify the fraction, divide both the numerator and the denominator by the GCD, which is 125.

$\frac{125}{1000} = \frac{125 \div 125}{1000 \div 125} = \frac{1}{8}$

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

A: To evaluate the cube root of a fraction, take the cube root of the numerator and the cube root of the denominator.

$\sqrt[3]{\frac{1}{8}} = \frac{\sqrt[3]{1}}{\sqrt[3]{8}}$

Q: How do I simplify the cube root of 8?

A: The cube root of 8 can be simplified by finding the cube root of 2, which is 2.

$\frac{\sqrt[3]{1}}{\sqrt[3]{8}} = \frac{1}{2}$

Q: How do I convert a fraction to a decimal?

A: To convert a fraction to a decimal, divide the numerator by the denominator.

$\frac{1}{2} = 0.5$

Q: How do I compare my answer to the answer choices?

A: To compare your answer to the answer choices, convert each of the answer choices to a decimal.

A. $\frac{25}{200} = 0.125$

B. $\frac{15}{100} = 0.15$

C. $\frac{3}{25} = 0.12$

D. $\frac{5}{10} = 0.5$

Q: What is the correct answer?

A: The correct answer is D. $\frac{5}{10}$, which is equivalent to 0.5.

Common Mistakes

Mistake 1: Not simplifying the fraction

• Not simplifying the fraction can lead to incorrect answers.
• Make sure to simplify the fraction by dividing both the numerator and the denominator by the GCD.

Mistake 2: Not evaluating the cube root correctly

• Not evaluating the cube root correctly can lead to incorrect answers.
• Make sure to take the cube root of the numerator and the cube root of the denominator.

Mistake 3: Not converting the fraction to a decimal

• Not converting the fraction to a decimal can lead to incorrect answers.
• Make sure to convert the fraction to a decimal by dividing the numerator by the denominator.

Tips and Tricks

Tip 1: Simplify the fraction first

• Simplifying the fraction first can make it easier to evaluate the cube root.
• Make sure to divide both the numerator and the denominator by the GCD.

Tip 2: Evaluate the cube root correctly

• Evaluating the cube root correctly is crucial to getting the correct answer.
• Make sure to take the cube root of the numerator and the cube root of the denominator.

Tip 3: Convert the fraction to a decimal

• Converting the fraction to a decimal can make it easier to compare the answer to the answer choices.
• Make sure to divide the numerator by the denominator.

Conclusion

Evaluating the expression $\sqrt[3]{\frac{125}{1000}}$ requires simplifying the fraction, evaluating the cube root, and converting the fraction to a decimal. By following these steps and avoiding common mistakes, you can get the correct answer.