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

Introduction

Simplifying complex expressions is a crucial skill in mathematics, and it requires careful attention to detail. In this article, we will analyze a given expression and identify the mistake made by Donte in simplifying it. We will also discuss the correct steps to simplify the expression and provide a clear understanding of the distributive property.

The Expression

The given expression is:

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

Donte's Mistake

Donte's simplified expression is:

$$-4 + 17i$$

Analysis of Donte's Mistake

To identify the mistake made by Donte, let's analyze the steps he took to simplify the expression.

Step 1: Distributive Property

The first step in simplifying the expression is to apply the distributive property to the term $4(1+3i)$. The distributive property states that for any real numbers $a$, $b$, and $c$, the following equation holds:

$$a(b+c) = ab + ac$$

In this case, we have:

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

Using the distributive property, we get:

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

Step 2: Subtracting the Second Term

The next step is to subtract the second term $(8-5i)$ from the result obtained in the previous step.

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

To subtract the second term, we need to distribute the negative sign to both terms inside the parentheses.

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

Step 3: Combining Like Terms

The final step is to combine like terms.

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

Combining like terms, we get:

$$-4 + 17i$$

Correct Simplification

Now, let's analyze the correct steps to simplify the expression.

Step 1: Distributive Property

The first step is to apply the distributive property to the term $4(1+3i)$.

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

Using the distributive property, we get:

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

Step 2: Subtracting the Second Term

The next step is to subtract the second term $(8-5i)$ from the result obtained in the previous step.

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

To subtract the second term, we need to distribute the negative sign to both terms inside the parentheses.

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

Step 3: Combining Like Terms

The final step is to combine like terms.

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

Combining like terms, we get:

$$-4 + 17i$$

Conclusion

In this article, we analyzed a given expression and identified the mistake made by Donte in simplifying it. We also discussed the correct steps to simplify the expression and provided a clear understanding of the distributive property. The correct simplified expression is indeed $-4 + 17i$, but the mistake made by Donte was not in the simplification process itself, but rather in the fact that he did not apply the distributive property correctly for the term $4(1+3i)$.

Final Answer

The final answer is A. He did not apply the distributive property correctly for $4(1+3i)$.

Introduction

In our previous article, we analyzed a given expression and identified the mistake made by Donte in simplifying it. We also discussed the correct steps to simplify the expression and provided a clear understanding of the distributive property. In this article, we will provide a Q&A section to further clarify any doubts and provide additional information on the topic.

Q&A

Q1: What is the distributive property, and how is it used in simplifying expressions?

A1: The distributive property is a mathematical concept that states that for any real numbers $a$, $b$, and $c$, the following equation holds:

$$a(b+c) = ab + ac$$

In the context of simplifying expressions, the distributive property is used to expand the product of a number and a binomial (a sum of two terms).

Q2: How do you apply the distributive property to the term $4(1+3i)$?

A2: To apply the distributive property to the term $4(1+3i)$, we need to multiply the number $4$ by each term inside the parentheses.

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

Using the distributive property, we get:

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

Q3: What is the mistake made by Donte in simplifying the expression?

A3: The mistake made by Donte is not in the simplification process itself, but rather in the fact that he did not apply the distributive property correctly for the term $4(1+3i)$.

Q4: How do you subtract the second term $(8-5i)$ from the result obtained in the previous step?

A4: To subtract the second term $(8-5i)$ from the result obtained in the previous step, we need to distribute the negative sign to both terms inside the parentheses.

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

Using the distributive property, we get:

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

Q5: How do you combine like terms in the expression?

A5: To combine like terms in the expression, we need to add or subtract the coefficients of the like terms.

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

Combining like terms, we get:

$$-4 + 17i$$

Q6: What is the final simplified expression?

A6: The final simplified expression is:

$$-4 + 17i$$

Conclusion

In this Q&A article, we provided additional information and clarification on the topic of simplifying expressions and the distributive property. We hope that this article has been helpful in understanding the concept and has provided a clear understanding of the mistake made by Donte in simplifying the expression.

Final Answer

The final answer is A. He did not apply the distributive property correctly for $4(1+3i)$.