Select All The Correct Answers.Which Expressions Are Equivalent To The Given Expression? ( − 9 + − 4 ) − ( 2 576 + − 64 (-\sqrt{9}+\sqrt{-4})-(2 \sqrt{576}+\sqrt{-64} ( − 9 ​ + − 4 ​ ) − ( 2 576 ​ + − 64 ​ ]A. − 3 + 2 I + 2 ( 24 ) + 8 I -3 + 2i + 2(24) + 8i − 3 + 2 I + 2 ( 24 ) + 8 I B. − 3 − 2 I − 2 ( 24 ) + 8 I -3 - 2i - 2(24) + 8i − 3 − 2 I − 2 ( 24 ) + 8 I C. − 51 − 6 I -51 - 6i − 51 − 6 I D. $-51 +

Understanding the Given Expression

The given expression is $(-\sqrt{9}+\sqrt{-4})-(2 \sqrt{576}+\sqrt{-64})$. To simplify this expression, we need to evaluate the square roots and then perform the operations.

Evaluating Square Roots

• $\sqrt{9}$ is equal to $3$ because $3^2 = 9$.
• $\sqrt{-4}$ is equal to $2i$ because $i^2 = -1$ and $2^2 = 4$.
• $\sqrt{576}$ is equal to $24$ because $24^2 = 576$.
• $\sqrt{-64}$ is equal to $8i$ because $i^2 = -1$ and $8^2 = 64$.

Substituting the Values

Now that we have evaluated the square roots, we can substitute the values back into the original expression:

$(-\sqrt{9}+\sqrt{-4})-(2 \sqrt{576}+\sqrt{-64})$

$= (-3 + 2i) - (2(24) + 8i)$

Simplifying the Expression

To simplify the expression, we need to distribute the negative sign to the terms inside the parentheses:

$= -3 + 2i - 48 - 8i$

Combining Like Terms

Now, we can combine the like terms:

$= -3 - 48 + 2i - 8i$

$= -51 - 6i$

Conclusion

The simplified expression is $-51 - 6i$. Therefore, the correct answer is:

C. $-51 - 6i$

Discussion

This problem requires the application of mathematical concepts such as square roots, distribution, and combining like terms. The correct answer can be obtained by carefully evaluating the square roots and then simplifying the expression.

Key Takeaways

• To simplify complex expressions, we need to evaluate the square roots and then perform the operations.
• We can use the properties of square roots, such as $\sqrt{a^2} = a$, to simplify the expression.
• We can also use the distributive property to distribute the negative sign to the terms inside the parentheses.
• Finally, we can combine the like terms to simplify the expression.

Practice Problems

1. Simplify the expression: $(\sqrt{16} + \sqrt{-9}) - (2 \sqrt{225} + \sqrt{-100})$
2. Simplify the expression: $(\sqrt{25} - \sqrt{-36}) - (2 \sqrt{49} + \sqrt{-121})$
3. Simplify the expression: $(\sqrt{81} + \sqrt{-4}) - (2 \sqrt{144} + \sqrt{-64})$

Answer Key

1. $-5 - 4i$
2. $-7 + 11i$
3. $-51 - 6i$
   Simplifying Complex Expressions: A Q&A Guide =====================================================

Q: What is the first step in simplifying a complex expression?

A: The first step in simplifying a complex expression is to evaluate the square roots. This involves finding the square root of each number and simplifying the expression.

Q: How do I simplify a square root of a negative number?

A: To simplify a square root of a negative number, you can use the property of square roots that states $\sqrt{-a} = i\sqrt{a}$. For example, $\sqrt{-4} = i\sqrt{4} = 2i$.

Q: What is the distributive property, and how is it used in simplifying complex expressions?

A: The distributive property is a mathematical property that states $a(b + c) = ab + ac$. In simplifying complex expressions, the distributive property is used to distribute the negative sign to the terms inside the parentheses.

Q: How do I combine like terms in a complex expression?

A: To combine like terms in a complex expression, you need to identify the like terms and add or subtract their coefficients. For example, in the expression $-3 + 2i - 48 - 8i$, the like terms are $-3$ and $-48$, and the like terms are $2i$ and $-8i$. Combining these terms gives $-51 - 6i$.

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

A: Some common mistakes to avoid when simplifying complex expressions include:

• Not evaluating the square roots correctly
• Not using the distributive property correctly
• Not combining like terms correctly
• Not checking the final answer for errors

Q: How can I practice simplifying complex expressions?

A: You can practice simplifying complex expressions by working through practice problems, such as the ones listed below:

1. Simplify the expression: $(\sqrt{16} + \sqrt{-9}) - (2 \sqrt{225} + \sqrt{-100})$
2. Simplify the expression: $(\sqrt{25} - \sqrt{-36}) - (2 \sqrt{49} + \sqrt{-121})$
3. Simplify the expression: $(\sqrt{81} + \sqrt{-4}) - (2 \sqrt{144} + \sqrt{-64})$

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

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

• Calculating the area and perimeter of complex shapes
• Solving systems of equations
• Modeling population growth and decay
• Calculating the trajectory of projectiles

Q: How can I use technology to simplify complex expressions?

A: You can use technology, such as calculators or computer software, to simplify complex expressions. For example, you can use a calculator to evaluate the square roots and then simplify the expression.

Q: What are some common errors to look out for when simplifying complex expressions?

A: Some common errors to look out for when simplifying complex expressions include:

• Not evaluating the square roots correctly
• Not using the distributive property correctly
• Not combining like terms correctly
• Not checking the final answer for errors

Conclusion

Simplifying complex expressions is an important skill in mathematics that has many real-world applications. By following the steps outlined in this guide, you can simplify complex expressions and avoid common mistakes. Remember to practice regularly and use technology to help you simplify complex expressions.