In Exercises 9–20, find each product and write the result in s...

Question

In Exercises 9–20, find each product and write the result in standard form. (−5 + i)(−5 − i)

Accepted Answer

Welcome back. Everyone. In this problem, we want to multiply negative 13 plus I by negative 13 minus I and express our number in standard form if possible for our answer choices A is negative 169 plus IB is 169 minus IC is D is negative 170. Now, what do we know here? Well, we want to multiply, let me write a to negative 13 plus I by negative 13 minus I. And when we do it, we want to express our answer in standard form. Recall that the standard form for complex numbers means that we write it in the form A plus B I where A is a real number and B is an imaginary number. So if we go ahead and multiply, then we should be able to simplify and write it in standard form. Now, I want you to look very closely at the product here, negative 13 plus I multiplied by negative 13 minus I, doesn't it look like an algebraic property you've seen before? Doesn't this look like the difference of two squares? Remember again that the difference of two squares tells us that if we are multiplying the sum and the difference of two expressions, it equals the difference of their squares. In other words, the square of one expression minus the square of the other. Notice both of them have negative 13 and I where one is the difference and one is the sum. So by applying that property, we can rewrite our expression as negative squared minus I squared. No, we can go from here. Again, there's another property that can help us another property of complex numbers because recall that I squared equals one. That's a property or a negative one. Rather I squared equals negative one. That's a property of complex numbers just to jog your memory a little bit. That's because I equals the square root of negative one. So I squared would have been the square root of negative one R squared, which would just give us negative one. So by applying that property, we can rewrite our expression as negative 13 squared. Well, negative 13 squared is 169 and I would be subtracting I squared which is negative one, 1 69 minus negative one becomes 1 69 plus one which equals 1 70. So here there is no imaginary part for a number, but we have a real part. So it's in standard form. Therefore, our answer would have been answer choice. C 170. Thanks a lot for watching everyone. I hope this video helped

In Exercises 9–20, find each product and write the result in standard form. (−5 + i)(−5 − i)

Key Concepts

Complex Numbers and Standard Form

Multiplication of Complex Numbers

Imaginary Unit Properties

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