Daine's Mistake In Simplifying 8(1+2i)-(7-3i) Complex Expression
Daine attempted to simplify the complex expression $8(1+2i)-(7-3i)$ and arrived at the result $1+5i$. However, this result is incorrect, indicating a mistake in the simplification process. To identify Daine's error, we need to break down the steps involved in simplifying the expression and compare them to Daine's work.
Understanding the Problem
The expression involves complex numbers, which are numbers of the form $a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit, defined as $i^2 = -1$. The expression requires us to perform two main operations:
- Distribution: Multiplying the term outside the parentheses by each term inside the parentheses.
- Combining Like Terms: Adding or subtracting the real and imaginary parts separately.
Let's go through the correct steps to simplify the expression.
Correct Simplification Steps
Step 1: Distribute the 8
The first part of the expression is $8(1+2i)$. We need to distribute the 8 to both the real part (1) and the imaginary part ($2i$):
This step involves multiplying 8 by both 1 and $2i$. The result should be $8 + 16i$. Making sure to correctly apply the distributive property is crucial here. This is where many mistakes can happen, so let's keep this in mind.
Step 2: Distribute the Subtraction Sign
The second part of the expression is $-(7-3i)$. We need to distribute the negative sign (which is equivalent to multiplying by -1) to both terms inside the parentheses:
Distributing the subtraction sign correctly means changing the sign of each term inside the parentheses. So, 7 becomes -7, and -3i becomes +3i. This is another critical step where errors often occur. Getting the signs right is super important, guys!
Step 3: Combine Like Terms
Now we combine the results from the previous two steps:
To combine like terms, we add the real parts together and the imaginary parts together:
- Real parts: $8 + (-7) = 1$
- Imaginary parts: $16i + 3i = 19i$
So, the simplified expression is $1 + 19i$. This involves adding the real parts (8 and -7) and the imaginary parts (16i and 3i) separately. We end up with $1 + 19i$. This is the final simplified form of the expression.
Step 4: Compare with Daine's Result
Daine's result was $1 + 5i$, but the correct simplified expression is $1 + 19i$. Clearly, Daine made a mistake somewhere in the process. Let's analyze the possible errors based on the options provided.
Analyzing the Possible Mistakes
We need to figure out what went wrong in Daine's simplification. The two main areas where mistakes usually happen in this type of problem are:
- Distributing the initial multiplication.
- Distributing the negative sign in the subtraction.
Let's look at each potential mistake more closely.
Option A: Incorrect Distribution of $8(1+2i)$
If Daine did not apply the distributive property correctly for $8(1+2i)$, it means he might have made a mistake in multiplying 8 by both 1 and $2i$. The correct distribution should yield:
If Daine got something different, like $8 + 2i$ or $8 + 10i$, then this would be the error. This is a common mistake, so it’s worth investigating.
Option B: Incorrect Distribution of the Subtraction Sign
If Daine did not distribute the subtraction sign correctly for $-(7-3i)$, he might have failed to change the signs of both terms inside the parentheses. The correct distribution should yield:
If Daine instead wrote $-7 - 3i$, this would indicate a failure to correctly distribute the negative sign. This is another frequent error, so we need to consider this as a strong possibility.
Identifying Daine's Mistake
To pinpoint Daine's mistake, let's consider what the result $1 + 5i$ implies. If Daine’s final result was $1 + 5i$, we can reverse-engineer the steps to see where the error might have occurred.
The correct expression is:
Let's assume Daine correctly distributed the 8:
If the final result was $1 + 5i$, then the error likely occurred when distributing the subtraction sign. Suppose Daine incorrectly distributed the negative sign:
Combining like terms here would give:
This result isn't $1 + 5i$, so the mistake isn't solely in the subtraction sign distribution. Now, let's explore the possibility that Daine made a mistake in the initial distribution of 8. If Daine had, for example, calculated $8(1+2i)$ as $8 + 2i$ (incorrectly), then the expression would look like this:
Distributing the subtraction sign correctly here:
Combining like terms:
Aha! This matches Daine's incorrect result. Therefore, Daine's mistake was most likely in the initial distribution.
Conclusion
Based on our analysis, Daine's mistake was A. He did not apply the distributive property correctly for $8(1+2i)$. By incorrectly calculating $8(1+2i)$ as something other than $8 + 16i$, Daine arrived at the wrong final answer. It's super important to double-check those distribution steps, guys! Always make sure you're multiplying correctly and paying attention to the signs.
This exercise highlights the importance of careful, step-by-step simplification and the critical role of the distributive property in algebraic manipulations. Keep practicing, and you'll nail these problems every time!
