Complex Number Operations: Find True Statements

Hey math enthusiasts! Let's dive into the fascinating world of complex numbers and tackle a problem that involves basic operations. We'll be working with complex numbers in the form of $a + bi$, where 'a' is the real part and 'b' is the imaginary part, and 'i' is the imaginary unit ($i^2 = -1$). Today, we'll focus on determining the correct results of complex number multiplication and division. So, let's sharpen our pencils and get started!

Problem Setup: Defining Complex Numbers a and b

Before we begin, let's define our complex numbers. We're given two complex numbers:

• $a = 3 - 5i$
• $b = -1 + 7i$

The goal here is, guys, to figure out which of the given statements involving the product ($ab$) and quotient ($\frac{a}{b}$) of these numbers are actually true. So, we're not just doing calculations; we're also acting like detectives, checking which solutions match our findings. How cool is that?

Calculating the Product: $ab$

First, let's calculate the product $ab$. Remember, when we multiply complex numbers, we use the distributive property, just like with regular binomials. We also need to keep in mind that $i^2 = -1$. Let's break it down:

$ab = (3 - 5i)(-1 + 7i)$

Now, we'll expand this using the FOIL method (First, Outer, Inner, Last):

• First: $3 \times -1 = -3$
• Outer: $3 \times 7i = 21i$
• Inner: $-5i \times -1 = 5i$
• Last: $-5i \times 7i = -35i^2$

So, we have:

$ab = -3 + 21i + 5i - 35i^2$

Now, let's substitute $i^2$ with $-1$:

$ab = -3 + 21i + 5i - 35(-1)$

Which simplifies to:

$ab = -3 + 21i + 5i + 35$

Combine the real and imaginary parts:

$ab = (-3 + 35) + (21i + 5i)$

Finally, we get:

$ab = 32 + 26i$

So, the correct product of $a$ and $b$ is $32 + 26i$. This means the statement "$ab = 32 + 26i$" is true, while the other options for $ab$ are incorrect. We're on a roll!

Calculating the Quotient: $\frac{a}{b}$

Now, let's tackle the division of complex numbers, which is a bit trickier but totally manageable. To divide complex numbers, we need to multiply both the numerator and the denominator by the conjugate of the denominator. Remember, the complex conjugate of a complex number $a + bi$ is $a - bi$. It’s like its twin, but with the opposite sign for the imaginary part. This helps us get rid of the imaginary part in the denominator.

So, for $b = -1 + 7i$, the conjugate is $-1 - 7i$. Let’s set up our division:

$\frac{a}{b} = \frac{3 - 5i}{-1 + 7i}$

Now, we'll multiply both the numerator and denominator by the conjugate of the denominator:

$\frac{a}{b} = \frac{(3 - 5i)(-1 - 7i)}{(-1 + 7i)(-1 - 7i)}$

Let's handle the numerator first. We'll use the FOIL method again:

$(3 - 5i)(-1 - 7i) =$

• First: $3 \times -1 = -3$
• Outer: $3 \times -7i = -21i$
• Inner: $-5i \times -1 = +5i$
• Last: $-5i \times -7i = 35i^2$

So, the numerator expands to:

$-3 - 21i + 5i + 35i^2$

Substitute $i^2$ with $-1$:

$-3 - 21i + 5i + 35(-1) = -3 - 21i + 5i - 35$

Combine the real and imaginary parts:

$(-3 - 35) + (-21i + 5i) = -38 - 16i$

Now, let's deal with the denominator:

$(-1 + 7i)(-1 - 7i) =$

• First: $-1 \times -1 = 1$
• Outer: $-1 \times -7i = 7i$
• Inner: $7i \times -1 = -7i$
• Last: $7i \times -7i = -49i^2$

So, the denominator expands to:

$1 + 7i - 7i - 49i^2$

Notice that the imaginary terms ($7i$ and $-7i$) cancel each other out, which is exactly what we want when we use conjugates. Now, substitute $i^2$ with $-1$:

$1 - 49(-1) = 1 + 49 = 50$

So, our quotient $\frac{a}{b}$ becomes:

$\frac{a}{b} = \frac{-38 - 16i}{50}$

Now, let's simplify by dividing both the real and imaginary parts by their greatest common divisor, which is 2:

$\frac{a}{b} = \frac{-19 - 8i}{25}$

We can also write this as:

$\frac{a}{b} = -\frac{19 + 8i}{25}$

Comparing this result with the given statements, we find that the statement "$\frac{a}{b} = -\frac{19 + 8i}{25}$" is true. Woo-hoo! We nailed it!

Identifying the True Statements: Our Final Answer

Alright, folks, we've done the heavy lifting! We calculated both the product and the quotient of the complex numbers $a$ and $b$. Now, let's summarize our findings and identify the true statements:

• We found that $ab = 32 + 26i$, so the statement "$ab = 32 + 26i$" is true.
• We also found that $\frac{a}{b} = -\frac{19 + 8i}{25}$, so the statement "$\frac{a}{b} = -\frac{19 + 8i}{25}$" is true.

Therefore, the true statements are:

• $ab = 32 + 26i$
• $\frac{a}{b} = -\frac{19 + 8i}{25}$

Conclusion: Mastering Complex Number Operations

Great job, everyone! We've successfully navigated the world of complex number operations and identified the true statements. Remember, the key to success with complex numbers is to treat them like binomials when multiplying, and to use conjugates when dividing. And most importantly, don't forget that $i^2 = -1$. Keep practicing, and you'll become a complex number whiz in no time!

I hope this breakdown was helpful and clear. Keep exploring the fascinating world of mathematics, and remember to have fun while you're at it! Until next time, happy calculating!