Donte Simplified The Expression Below:${ \begin{array}{l} 4(1+3i) - (8-5i) \ 4 + 12i - 8 + 5i \ -4 + 17i \end{array} }$What Mistake Did Donte Make?A. He Did Not Apply The Distributive Property Correctly For 4 ( 1 + 3 I 4(1+3i 4 ( 1 + 3 I ].B. He Did Not

Understanding Complex Expressions

Complex expressions are mathematical expressions that involve both real and imaginary numbers. In the given expression, Donte simplified the expression $4(1+3i) - (8-5i)$ to $-4 + 17i$. However, we need to examine the steps taken by Donte to identify any potential mistakes.

Step 1: Distributive Property

The distributive property states that for any real numbers $a$, $b$, and $c$, $a(b+c) = ab + ac$. In the given expression, Donte needs to apply the distributive property to expand the term $4(1+3i)$.

Applying the Distributive Property

To apply the distributive property, we multiply the real and imaginary parts of the term $1+3i$ by $4$. This gives us:

$$4(1+3i) = 4(1) + 4(3i) = 4 + 12i$$

Step 2: Subtracting the Second Term

Now that we have expanded the first term, we can proceed to subtract the second term $(8-5i)$ from the expanded first term $4 + 12i$.

$$4 + 12i - (8-5i) = 4 + 12i - 8 + 5i$$

Simplifying the Expression

To simplify the expression, we combine like terms. The real parts are $4$ and $-8$, which combine to give $-4$. The imaginary parts are $12i$ and $5i$, which combine to give $17i$.

$$4 + 12i - 8 + 5i = -4 + 17i$$

Identifying Donte's Mistake

However, upon closer inspection, we notice that Donte made a mistake in applying the distributive property. He correctly expanded the first term $4(1+3i)$ to $4 + 12i$, but he did not distribute the negative sign to the second term $(8-5i)$.

Correcting Donte's Mistake

To correct Donte's mistake, we need to distribute the negative sign to the second term $(8-5i)$. This gives us:

$$4(1+3i) - (8-5i) = 4 + 12i - 8 + 5i$$

Now, we can simplify the expression by combining like terms.

$$4 + 12i - 8 + 5i = -4 + 17i$$

Conclusion

In conclusion, Donte made a mistake in applying the distributive property to the second term $(8-5i)$. He did not distribute the negative sign to the second term, which resulted in an incorrect simplification of the expression. By correcting this mistake, we can simplify the expression correctly to $-4 + 17i$.

Common Mistakes in Simplifying Complex Expressions

When simplifying complex expressions, it is essential to apply the distributive property correctly. Here are some common mistakes to watch out for:

• Not distributing the negative sign: Failing to distribute the negative sign to a term can result in an incorrect simplification of the expression.
• Not combining like terms: Failing to combine like terms can result in an incorrect simplification of the expression.
• Not using the correct order of operations: Failing to use the correct order of operations (PEMDAS) can result in an incorrect simplification of the expression.

Tips for Simplifying Complex Expressions

When simplifying complex expressions, here are some tips to keep in mind:

• Apply the distributive property correctly: Make sure to distribute the negative sign to all terms.
• Combine like terms: Make sure to combine like terms to simplify the expression.
• Use the correct order of operations: Make sure to use the correct order of operations (PEMDAS) to simplify the expression.

Q: What is a complex expression?

A: A complex expression is a mathematical expression that involves both real and imaginary numbers. It can be a combination of numbers, variables, and mathematical operations.

Q: What is the distributive property?

A: The distributive property is a mathematical property that states that for any real numbers $a$, $b$, and $c$, $a(b+c) = ab + ac$. It allows us to expand a term that is being multiplied by a sum of two or more terms.

Q: How do I apply the distributive property to a complex expression?

A: To apply the distributive property, you need to multiply the real and imaginary parts of the term being multiplied by the sum of two or more terms. For example, if you have the expression $4(1+3i)$, you would multiply the real part $1$ by $4$ and the imaginary part $3i$ by $4$.

Q: What is the correct order of operations for simplifying complex expressions?

A: The correct order of operations for simplifying complex expressions is:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I simplify a complex expression with multiple terms?

A: To simplify a complex expression with multiple terms, you need to combine like terms. Like terms are terms that have the same variable or variables raised to the same power. For example, if you have the expression $4 + 12i - 8 + 5i$, you would combine the real parts $4$ and $-8$ to get $-4$, and combine the imaginary parts $12i$ and $5i$ to get $17i$.

Q: What are some common mistakes to watch out for when simplifying complex expressions?

A: Some common mistakes to watch out for when simplifying complex expressions include:

• Not distributing the negative sign: Failing to distribute the negative sign to a term can result in an incorrect simplification of the expression.
• Not combining like terms: Failing to combine like terms can result in an incorrect simplification of the expression.
• Not using the correct order of operations: Failing to use the correct order of operations (PEMDAS) can result in an incorrect simplification of the expression.

Q: How can I practice simplifying complex expressions?

A: You can practice simplifying complex expressions by working through examples and exercises in a math textbook or online resource. You can also try simplifying complex expressions on your own using a calculator or computer program.

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

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

• Engineering: Simplifying complex expressions is essential in engineering, where complex mathematical models are used to design and analyze systems.
• Physics: Simplifying complex expressions is essential in physics, where complex mathematical models are used to describe the behavior of physical systems.
• Computer Science: Simplifying complex expressions is essential in computer science, where complex mathematical models are used to develop algorithms and data structures.

By following these tips and avoiding common mistakes, you can simplify complex expressions correctly and accurately.