Video: Finding the Product of Two Complex Numbers in Algebraic Form

Expand and simplify (4 − 𝑖)(3 + 2𝑖).

03:39

Video Transcript

Expand and simplify four minus 𝑖 multiplied by three plus two 𝑖.

Okay, first look at this question, we can see that okay we’ve got four minus 𝑖 multiplied by three plus two 𝑖. Well we’re going to expand these parentheses in the same way we’d expand any parentheses. It doesn’t matter that we’ve got 𝑖 or an imaginary number. We’ll just carry it out in the normal way.

So if we have our two parenthesis, then the first thing we’re gonna multiply together is our first term of each. So we’re gonna have four multiplied by three, which is gonna give us 12. Next, we’re gonna multiply our four, cause that’s our first term in our first parenthesis, by two 𝑖, which is the second term in our second parenthesis.

So that’s gonna give us eight 𝑖. Then next, we’ve actually got the second term in our first parenthesis, which is negative 𝑖. Okay, be careful! Make sure we’re including the correct sign here. So it’s negative 𝑖. And that’s gonna be multiplied by three. So that’s gonna give us negative three 𝑖. And then finally, we’re gonna multiply the second term in each parenthesis.

So we’re gonna have negative 𝑖 multiplied by two 𝑖. And we’re gonna multiply this in the same way we would multiply any term normally. So we’re actually gonna get negative two 𝑖 squared. Okay, great! So we’ve actually expanded our parentheses. And we’ve got the four terms. Right! So now, same as always, we’re actually going to collect our terms and simplify.

So we’re gonna have 12 and then we got plus eight 𝑖 minus three 𝑖, which is gonna leave us with plus five 𝑖. And then we’ve got minus two 𝑖 squared. Okay, so great! We’ve reached this stage. We’ve got 12 plus five 𝑖 minus two 𝑖 squared. Well it’s actually at this point now that we actually do take in consideration the fact that we have imaginary numbers so we have 𝑖.

So what we want to do is we have to actually think about 𝑖 squared. So what does that actually mean? Well let’s go back to actually thinking about what 𝑖 means. Well what 𝑖 means is square root of negative one. Okay, so let’s use this. If we know that 𝑖 is equal to the square root of negative one, let’s go back and see what 𝑖 squared is going to equal.

Well if we actually think about what we’ve got, 𝑖 squared is gonna be equal to square root of negative one, because that’s 𝑖. And then that’s going to be all squared because, obviously, we’re squaring our 𝑖. So therefore, if we’re gonna square a square root, then this is gonna be equal to negative one. And if we think about why that is, we can think about it in two ways.

One, if we’re squaring a square root, we’re actually kind of reversing the process so we’re left with the number inside. But also if we think about one of our exponent rules, if we have 𝑎 to the power of 𝑚 all to the power of 𝑛 it’s going to be the same as 𝑎 to the power of 𝑚𝑛. In that case, what we’d have is negative one to power of a half multiplied by two. That means a half multiplied by two is one. So then, we’re just gonna have negative one.

Okay, great! So we now know what 𝑖 squared is. We can actually get on and simplify even further. So now what we’re gonna do is if we substitute 𝑖 squared for negative one, we’re going to get 12 plus five 𝑖 minus, and then two multiplied by negative one, which will be equal to 12 plus five 𝑖 minus negative two.

Well if we’re gonna minus negative two, this is the same as adding on two. So therefore, if we expand and simplify four minus 𝑖 multiplied by three plus two 𝑖, we’re gonna get 14 plus five 𝑖. And we got the 14 because had 12 plus two gives us 14, so 14 plus five 𝑖.

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