Simplify 25 100 \sqrt{\frac{25}{100}} 100 25 ​ ​ .

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Understanding the Problem

When dealing with square roots of fractions, it's essential to simplify the fraction first before finding the square root. In this case, we have the expression $\sqrt{\frac{25}{100}}$. Our goal is to simplify this expression by finding the square root of the numerator and the denominator separately.

Simplifying the Fraction

To simplify the fraction, we need to find the greatest common divisor (GCD) of the numerator and the denominator. The GCD of 25 and 100 is 25. We can divide both the numerator and the denominator by 25 to simplify the fraction.

$\frac{25}{100} = \frac{25 \div 25}{100 \div 25} = \frac{1}{4}$

Finding the Square Root

Now that we have simplified the fraction, we can find the square root of the numerator and the denominator separately. The square root of 1 is 1, and the square root of 4 is 2.

$\sqrt{\frac{1}{4}} = \frac{\sqrt{1}}{\sqrt{4}} = \frac{1}{2}$

Conclusion

In conclusion, we have simplified the expression $\sqrt{\frac{25}{100}}$ by finding the square root of the numerator and the denominator separately. We first simplified the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 25. Then, we found the square root of the simplified fraction, which is $\frac{1}{2}$.

Real-World Applications

Simplifying square roots of fractions is an essential skill in mathematics, and it has numerous real-world applications. For example, in physics, we often encounter expressions involving square roots of fractions when dealing with wave functions and probability amplitudes. In engineering, we use square roots of fractions to calculate stress and strain on materials.

Tips and Tricks

When simplifying square roots of fractions, it's essential to remember the following tips and tricks:

• Always simplify the fraction first before finding the square root.
• Find the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by their GCD to simplify the fraction.
• Find the square root of the simplified fraction separately.

Common Mistakes

When simplifying square roots of fractions, it's easy to make mistakes. Here are some common mistakes to avoid:

• Not simplifying the fraction first before finding the square root.
• Not finding the greatest common divisor (GCD) of the numerator and the denominator.
• Not dividing both the numerator and the denominator by their GCD to simplify the fraction.
• Not finding the square root of the simplified fraction separately.

Practice Problems

Here are some practice problems to help you master the skill of simplifying square roots of fractions:

• Simplify $\sqrt{\frac{36}{144}}$.
• Simplify $\sqrt{\frac{49}{196}}$.
• Simplify $\sqrt{\frac{64}{256}}$.

Solutions

Here are the solutions to the practice problems:

• $\sqrt{\frac{36}{144}} = \frac{\sqrt{36}}{\sqrt{144}} = \frac{6}{12} = \frac{1}{2}$
• $\sqrt{\frac{49}{196}} = \frac{\sqrt{49}}{\sqrt{196}} = \frac{7}{14} = \frac{1}{2}$
• $\sqrt{\frac{64}{256}} = \frac{\sqrt{64}}{\sqrt{256}} = \frac{8}{16} = \frac{1}{2}$

Conclusion

In conclusion, simplifying square roots of fractions is an essential skill in mathematics, and it has numerous real-world applications. By following the tips and tricks outlined in this article, you can master the skill of simplifying square roots of fractions and solve complex problems with ease. Remember to always simplify the fraction first before finding the square root, and find the greatest common divisor (GCD) of the numerator and the denominator. With practice and patience, you can become a master of simplifying square roots of fractions.

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Frequently Asked Questions

Q: What is the greatest common divisor (GCD) of 25 and 100?

A: The greatest common divisor (GCD) of 25 and 100 is 25.

Q: How do I simplify the fraction $\frac{25}{100}$?

A: To simplify the fraction, you need to divide both the numerator and the denominator by their greatest common divisor, which is 25. $\frac{25}{100} = \frac{25 \div 25}{100 \div 25} = \frac{1}{4}$

Q: What is the square root of 1?

A: The square root of 1 is 1.

Q: What is the square root of 4?

A: The square root of 4 is 2.

Q: How do I find the square root of a fraction?

A: To find the square root of a fraction, you need to find the square root of the numerator and the denominator separately. $\sqrt{\frac{1}{4}} = \frac{\sqrt{1}}{\sqrt{4}} = \frac{1}{2}$

Q: What is the final answer to the expression $\sqrt{\frac{25}{100}}$?

A: The final answer to the expression $\sqrt{\frac{25}{100}}$ is $\frac{1}{2}$.

Q: Can I use a calculator to simplify the expression $\sqrt{\frac{25}{100}}$?

A: Yes, you can use a calculator to simplify the expression $\sqrt{\frac{25}{100}}$. However, it's essential to understand the underlying math to ensure accuracy.

Q: How do I apply this skill in real-world scenarios?

A: This skill is essential in various real-world scenarios, such as physics, engineering, and finance. For example, in physics, you may encounter expressions involving square roots of fractions when dealing with wave functions and probability amplitudes.

Q: What are some common mistakes to avoid when simplifying square roots of fractions?

A: Some common mistakes to avoid include not simplifying the fraction first, not finding the greatest common divisor (GCD) of the numerator and the denominator, and not dividing both the numerator and the denominator by their GCD to simplify the fraction.

Q: Can I use this skill to simplify other types of expressions?

A: Yes, you can use this skill to simplify other types of expressions involving square roots and fractions. However, you need to understand the underlying math and apply the correct techniques.

Q: How can I practice and improve my skills in simplifying square roots of fractions?

A: You can practice and improve your skills by working on more complex problems, using online resources, and seeking help from a teacher or tutor.

Additional Resources

• Mathway: An online math problem solver that can help you simplify square roots of fractions.
• Khan Academy: A free online learning platform that offers video lessons and practice exercises on simplifying square roots of fractions.
• Wolfram Alpha: A powerful online calculator that can help you simplify square roots of fractions and other mathematical expressions.

Conclusion

In conclusion, simplifying square roots of fractions is an essential skill in mathematics, and it has numerous real-world applications. By following the tips and tricks outlined in this article, you can master the skill of simplifying square roots of fractions and solve complex problems with ease. Remember to always simplify the fraction first before finding the square root, and find the greatest common divisor (GCD) of the numerator and the denominator. With practice and patience, you can become a master of simplifying square roots of fractions.