Simplify: $\sqrt{72 N^7}$.Provide Your Answer Below:

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Understanding the Problem

When simplifying a square root expression, we need to identify perfect square factors within the expression. In this case, we are given the expression $\sqrt{72 n^7}$, and our goal is to simplify it.

Breaking Down the Expression

To simplify the expression, we need to break it down into its prime factors. The expression $72 n^7$ can be written as $2^3 \cdot 3^2 \cdot n^7$. Now, we can rewrite the expression as $\sqrt{2^3 \cdot 3^2 \cdot n^7}$.

Identifying Perfect Square Factors

A perfect square factor is a factor that can be expressed as the square of an integer. In this case, we have $2^3$ and $3^2$ as perfect square factors. We can rewrite the expression as $\sqrt{(2^2)^2 \cdot 2 \cdot 3^2 \cdot n^7}$.

Simplifying the Expression

Now that we have identified the perfect square factors, we can simplify the expression. We can rewrite the expression as $\sqrt{(2^2)^2 \cdot 3^2 \cdot n^7}$. Using the property of square roots, we can rewrite the expression as $2^2 \cdot 3 \cdot n^3 \cdot \sqrt{2 \cdot n^4}$.

Final Simplification

The expression $2^2 \cdot 3 \cdot n^3 \cdot \sqrt{2 \cdot n^4}$ can be further simplified by combining like terms. We can rewrite the expression as $12 n^3 \sqrt{2 n^4}$.

Conclusion

In conclusion, the simplified expression for $\sqrt{72 n^7}$ is $12 n^3 \sqrt{2 n^4}$.

Step-by-Step Solution

Here is a step-by-step solution to the problem:

1. Break down the expression $72 n^7$ into its prime factors: $2^3 \cdot 3^2 \cdot n^7$.
2. Rewrite the expression as $\sqrt{2^3 \cdot 3^2 \cdot n^7}$.
3. Identify the perfect square factors: $2^2$ and $3^2$.
4. Rewrite the expression as $\sqrt{(2^2)^2 \cdot 2 \cdot 3^2 \cdot n^7}$.
5. Simplify the expression using the property of square roots: $2^2 \cdot 3 \cdot n^3 \cdot \sqrt{2 \cdot n^4}$.
6. Combine like terms to get the final simplified expression: $12 n^3 \sqrt{2 n^4}$.

Common Mistakes to Avoid

When simplifying square root expressions, it's easy to make mistakes. Here are some common mistakes to avoid:

• Not identifying perfect square factors
• Not using the property of square roots to simplify the expression
• Not combining like terms to get the final simplified expression

Real-World Applications

Simplifying square root expressions has many real-world applications. For example, in physics, we often need to simplify expressions involving square roots to solve problems involving motion and energy. In engineering, we need to simplify expressions involving square roots to design and build structures that can withstand various types of loads.

Practice Problems

Here are some practice problems to help you practice simplifying square root expressions:

1. Simplify $\sqrt{16 x^8}$.
2. Simplify $\sqrt{9 y^6}$.
3. Simplify $\sqrt{25 z^9}$.

Conclusion

In conclusion, simplifying square root expressions is an important skill that has many real-world applications. By following the steps outlined in this article, you can simplify even the most complex square root expressions. Remember to identify perfect square factors, use the property of square roots to simplify the expression, and combine like terms to get the final simplified expression.

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Frequently Asked Questions

Q: What is the simplified form of $\sqrt{72 n^7}$?

A: The simplified form of $\sqrt{72 n^7}$ is $12 n^3 \sqrt{2 n^4}$.

Q: How do I simplify a square root expression?

A: To simplify a square root expression, you need to identify perfect square factors within the expression. You can then use the property of square roots to simplify the expression.

Q: What are perfect square factors?

A: Perfect square factors are factors that can be expressed as the square of an integer. For example, $2^2$ is a perfect square factor because it can be expressed as $(2)^2$.

Q: How do I identify perfect square factors?

A: To identify perfect square factors, you need to break down the expression into its prime factors. You can then look for factors that are perfect squares.

Q: What is the property of square roots?

A: The property of square roots states that $\sqrt{a^2} = a$. This means that if you have a perfect square factor, you can simplify the expression by taking the square root of the factor.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the terms. For example, if you have the expression $2x + 3x$, you can combine the terms by adding the coefficients: $2x + 3x = 5x$.

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

A: Some common mistakes to avoid when simplifying square root expressions include:

• Not identifying perfect square factors
• Not using the property of square roots to simplify the expression
• Not combining like terms to get the final simplified expression

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

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

• Physics: Simplifying expressions involving square roots to solve problems involving motion and energy
• Engineering: Simplifying expressions involving square roots to design and build structures that can withstand various types of loads

Q: How can I practice simplifying square root expressions?

A: You can practice simplifying square root expressions by working through practice problems, such as:

• Simplify $\sqrt{16 x^8}$
• Simplify $\sqrt{9 y^6}$
• Simplify $\sqrt{25 z^9}$

Additional Resources

If you need additional help or resources to simplify square root expressions, here are some additional resources:

• Online tutorials and videos
• Practice problems and worksheets
• Online calculators and software
• Math textbooks and reference books

Conclusion

In conclusion, simplifying square root expressions is an important skill that has many real-world applications. By following the steps outlined in this article and practicing with additional resources, you can become proficient in simplifying square root expressions.

Common Misconceptions

Here are some common misconceptions about simplifying square root expressions:

• Misconception: Simplifying square root expressions is only for advanced math students. Reality: Simplifying square root expressions is a fundamental skill that can be learned by anyone with a basic understanding of math.
• Misconception: Simplifying square root expressions is only for specific types of expressions. Reality: Simplifying square root expressions can be applied to a wide range of expressions, including those with perfect square factors, rational expressions, and more.
• Misconception: Simplifying square root expressions is a difficult and time-consuming process. Reality: Simplifying square root expressions can be a straightforward and efficient process, especially with practice and experience.

Final Tips

Here are some final tips for simplifying square root expressions:

• Practice regularly to develop your skills and confidence.
• Use online resources and tools to help you simplify square root expressions.
• Break down complex expressions into smaller, more manageable parts.
• Use the property of square roots to simplify expressions.
• Combine like terms to get the final simplified expression.