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. (7 − 5i)(−2 − 3i)

Accepted Answer

Welcome back. Everyone. In this problem, we want to multiply four minus three, I by a negative five minus seven I. And we want to express our answer in standard form for our answer choices A is one plus 13. IB is 41 minus 43. IC is negative 41 minus 13 I and D is one minus 43. I. So in this problem, we're multiplying four minus three I by negative five minus seven, I, we can go ahead and multiply the expressions and then we can try to see if we can get it in standard form for complex numbers. That is, we want to write it in the form A plus B I where A is a real number and B is the imaginary number. So let's go ahead and multiply using the distributive property. We can rewrite this as four multiplied by negative five minus seven, I minus three. I multiplied by negative five minus seven, I four, multiplied by negative five is negative 24 multiplied by negative seven, I is negative 28 I and then negative three, I multiplied by negative five is a positive 15 I and negative three, I multiplied by negative seven, I would be positive 21 I squared. Now, before we go any further notice that we have an I squared term in our expression. And there is a property of complex numbers that tells us if you recall that I squared equals negative one. And just to jog your memory for that, remember that I by definition equals the spirit of negative one. So I squared would have been equal to the square root of negative one squared which is negative one. So we can use that property to help us simplify our expression. So by applying it, our expression is no negative 20 minus 28 I plus 15, I plus 21 multiplied by negative one. We can go ahead and simplify because no, we have like terms negative 2021 multiplied by negative one. A negative 20 plus 21 multiplied by negative one. I can write that here 21 multiplied by negative one is negative 21. And then we also have the like terms negative 28 I and 15 I and negative 28 I plus 15, I equals negative 13. I let's finally simplify, simplify our constant negative 20 minus ne sorry, negative 20 minus 21 is negative 41. So therefore our final answer is negative 41 minus 13. I, when we look in our answer choices that would have been answer choice. C 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. (7 − 5i)(−2 − 3i)

Key Concepts

Complex Number Multiplication

Standard Form of a Complex Number

Imaginary Unit Properties

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