Simplify $\sqrt{\frac{16}{49}}$.Be Sure To Write Your Answer In Simplest Form.

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Understanding the Problem

When simplifying a square root of a fraction, we need to simplify the fraction first and then find the square root of the numerator and denominator separately. In this case, we are given the expression $\sqrt{\frac{16}{49}}$ and we need to simplify it to its simplest form.

Breaking Down the Problem

To simplify the expression, we need to find the prime factors of both the numerator and the denominator. The numerator is 16 and the denominator is 49.

Prime Factors of 16

The prime factors of 16 are 2 x 2 x 2 x 2, or $2^4$.

Prime Factors of 49

The prime factors of 49 are 7 x 7, or $7^2$.

Simplifying the Fraction

Now that we have the prime factors of both the numerator and the denominator, we can simplify the fraction. We can rewrite the fraction as $\frac{2^4}{7^2}$.

Simplifying the Square Root

To simplify the square root, we can take the square root of the numerator and the denominator separately. The square root of $2^4$ is 2 x 2, or $2^2$. The square root of $7^2$ is 7.

Simplifying the Expression

Now that we have simplified the square root of the numerator and the denominator, we can simplify the expression. We can rewrite the expression as $\frac{2^2}{7}$.

Final Answer

The final answer is $\frac{4}{7}$.

Conclusion

Simplifying a square root of a fraction involves simplifying the fraction first and then finding the square root of the numerator and denominator separately. In this case, we simplified the expression $\sqrt{\frac{16}{49}}$ to its simplest form, which is $\frac{4}{7}$.

Additional Examples

Here are a few more examples of simplifying square roots of fractions:

• $\sqrt{\frac{25}{36}}$ = $\frac{5}{6}$
• $\sqrt{\frac{81}{100}}$ = $\frac{9}{10}$
• $\sqrt{\frac{144}{225}}$ = $\frac{12}{15}$

Tips and Tricks

When simplifying square roots of fractions, it's a good idea to start by simplifying the fraction first. This will make it easier to find the square root of the numerator and denominator separately. Additionally, make sure to check your work by plugging the simplified expression back into the original expression to make sure it's true.

Common Mistakes

One common mistake when simplifying square roots of fractions is to forget to simplify the fraction first. This can lead to incorrect answers. Another common mistake is to forget to check your work by plugging the simplified expression back into the original expression.

Real-World Applications

Simplifying square roots of fractions has many real-world applications. For example, in physics, we often need to simplify expressions involving square roots of fractions when working with equations of motion. In engineering, we often need to simplify expressions involving square roots of fractions when working with electrical circuits.

Final Thoughts

Simplifying square roots of fractions is an important skill to have in mathematics. By following the steps outlined in this article, you can simplify even the most complex expressions involving square roots of fractions. Remember to always simplify the fraction first and then find the square root of the numerator and denominator separately. With practice, you'll become a pro at simplifying square roots of fractions in no time!

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Frequently Asked Questions

Q: What is the first step in simplifying a square root of a fraction?

A: The first step in simplifying a square root of a fraction is to simplify the fraction itself. This involves finding the prime factors of both the numerator and the denominator.

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

A: To find the prime factors of a number, you need to break it down into its simplest building blocks. For example, the prime factors of 16 are 2 x 2 x 2 x 2, or $2^4$. The prime factors of 49 are 7 x 7, or $7^2$.

Q: What is the next step after simplifying the fraction?

A: After simplifying the fraction, the next step is to find the square root of the numerator and the denominator separately. This involves taking the square root of each prime factor.

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

A: To simplify the square root of a fraction, you need to take the square root of the numerator and the denominator separately. For example, the square root of $2^4$ is 2 x 2, or $2^2$. The square root of $7^2$ is 7.

Q: What is the final step in simplifying a square root of a fraction?

A: The final step in simplifying a square root of a fraction is to simplify the expression by combining the simplified square roots of the numerator and the denominator.

Q: Can you give me an example of simplifying a square root of a fraction?

A: Let's say we want to simplify the expression $\sqrt{\frac{25}{36}}$. To do this, we need to simplify the fraction first. The prime factors of 25 are 5 x 5, or $5^2$. The prime factors of 36 are 2 x 2 x 3 x 3, or $2^2 x 3^2$. We can rewrite the fraction as $\frac{5^2}{2^2 x 3^2}$. Taking the square root of each prime factor, we get $\frac{5}{2 x 3}$, which simplifies to $\frac{5}{6}$.

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

A: One common mistake to avoid is forgetting to simplify the fraction first. Another common mistake is forgetting to check your work by plugging the simplified expression back into the original expression.

Q: What are some real-world applications of simplifying square roots of fractions?

A: Simplifying square roots of fractions has many real-world applications. For example, in physics, we often need to simplify expressions involving square roots of fractions when working with equations of motion. In engineering, we often need to simplify expressions involving square roots of fractions when working with electrical circuits.

Q: How can I practice simplifying square roots of fractions?

A: You can practice simplifying square roots of fractions by working through examples and exercises. You can also try simplifying expressions involving square roots of fractions in real-world contexts, such as physics or engineering.

Q: What are some tips for simplifying square roots of fractions?

A: Here are some tips for simplifying square roots of fractions:

• Always simplify the fraction first.
• Take the square root of each prime factor.
• Simplify the expression by combining the simplified square roots of the numerator and the denominator.
• Check your work by plugging the simplified expression back into the original expression.

Conclusion

Simplifying square roots of fractions is an important skill to have in mathematics. By following the steps outlined in this article, you can simplify even the most complex expressions involving square roots of fractions. Remember to always simplify the fraction first and then find the square root of the numerator and denominator separately. With practice, you'll become a pro at simplifying square roots of fractions in no time!