Simplify: 7 20 7 \sqrt{20} 7 20 ​

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Understanding the Problem

When dealing with square roots, it's essential to simplify the expression to make it easier to work with. In this case, we're given the expression $7 \sqrt{20}$, and we need to simplify it. To start, let's break down the square root of 20 into its prime factors.

Breaking Down the Square Root

The square root of 20 can be written as $\sqrt{20}$. To simplify this, we need to find the prime factors of 20. The prime factorization of 20 is $2^2 \cdot 5$. So, we can rewrite the square root of 20 as $\sqrt{2^2 \cdot 5}$.

Simplifying the Square Root

Now that we have the prime factors of 20, we can simplify the square root. When we have a square root of a product, we can take the square root of each factor separately. In this case, we can take the square root of $2^2$ and the square root of 5. The square root of $2^2$ is 2, and the square root of 5 is $\sqrt{5}$.

Combining the Simplified Factors

Now that we have simplified the square root of 20, we can combine the simplified factors with the coefficient 7. The simplified expression is $7 \cdot 2 \cdot \sqrt{5}$.

Final Simplification

To simplify the expression further, we can combine the coefficient 7 with the factor 2. This gives us $14 \cdot \sqrt{5}$.

Conclusion

In conclusion, we have simplified the expression $7 \sqrt{20}$ to $14 \sqrt{5}$. This is the final simplified form of the expression.

Additional Tips and Tricks

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

• Break down the square root into its prime factors.
• Simplify each factor separately.
• Combine the simplified factors with the coefficient.
• Check your work to ensure that the expression is simplified correctly.

Real-World Applications

Simplifying square roots has many real-world applications, including:

• Calculating distances and lengths in geometry and trigonometry.
• Working with algebraic expressions and equations.
• Solving problems in physics and engineering.

Common Mistakes to Avoid

When simplifying square roots, it's essential to avoid the following common mistakes:

• Failing to break down the square root into its prime factors.
• Simplifying each factor incorrectly.
• Failing to combine the simplified factors with the coefficient.
• Not checking your work to ensure that the expression is simplified correctly.

Final Thoughts

Simplifying square roots is an essential skill in mathematics, and it has many real-world applications. By following the tips and tricks outlined in this article, you can simplify square roots with ease and confidence. Remember to break down the square root into its prime factors, simplify each factor separately, combine the simplified factors with the coefficient, and check your work to ensure that the expression is simplified correctly.

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Frequently Asked Questions

We've covered the basics of simplifying the expression $7 \sqrt{20}$, but we know that you may have some questions. Here are some frequently asked questions and answers to help you better understand the concept.

Q: What is the prime factorization of 20?

A: The prime factorization of 20 is $2^2 \cdot 5$.

Q: How do I simplify the square root of 20?

A: To simplify the square root of 20, you need to find the prime factors of 20. The prime factorization of 20 is $2^2 \cdot 5$. You can then take the square root of each factor separately. The square root of $2^2$ is 2, and the square root of 5 is $\sqrt{5}$.

Q: What is the simplified form of $7 \sqrt{20}$?

A: The simplified form of $7 \sqrt{20}$ is $14 \sqrt{5}$.

Q: How do I combine the simplified factors with the coefficient?

A: To combine the simplified factors with the coefficient, you need to multiply the coefficient with the simplified factors. In this case, you multiply 7 with 2 to get 14.

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

A: Some common mistakes to avoid when simplifying square roots include:

• Failing to break down the square root into its prime factors.
• Simplifying each factor incorrectly.
• Failing to combine the simplified factors with the coefficient.
• Not checking your work to ensure that the expression is simplified correctly.

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

A: Simplifying square roots has many real-world applications, including:

• Calculating distances and lengths in geometry and trigonometry.
• Working with algebraic expressions and equations.
• Solving problems in physics and engineering.

Q: How do I check my work to ensure that the expression is simplified correctly?

A: To check your work, you need to ensure that you have broken down the square root into its prime factors, simplified each factor correctly, combined the simplified factors with the coefficient, and checked your work to ensure that the expression is simplified correctly.

Q: What are some additional tips and tricks for simplifying square roots?

A: Some additional tips and tricks for simplifying square roots include:

• Breaking down the square root into its prime factors.
• Simplifying each factor separately.
• Combining the simplified factors with the coefficient.
• Checking your work to ensure that the expression is simplified correctly.

Conclusion

Simplifying square roots is an essential skill in mathematics, and it has many real-world applications. By following the tips and tricks outlined in this article, you can simplify square roots with ease and confidence. Remember to break down the square root into its prime factors, simplify each factor separately, combine the simplified factors with the coefficient, and check your work to ensure that the expression is simplified correctly.

Additional Resources

If you're looking for additional resources to help you learn more about simplifying square roots, here are some suggestions:

• Khan Academy: Simplifying Square Roots
• Mathway: Simplifying Square Roots
• Wolfram Alpha: Simplifying Square Roots

Final Thoughts

Simplifying square roots is a fundamental concept in mathematics, and it has many real-world applications. By following the tips and tricks outlined in this article, you can simplify square roots with ease and confidence. Remember to break down the square root into its prime factors, simplify each factor separately, combine the simplified factors with the coefficient, and check your work to ensure that the expression is simplified correctly.