In Exercises 1–10, perform the indicated operations and write the...

Question

In Exercises 1–10, perform the indicated operations and write the result in standard form.
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(−2 + √−100)²

Accepted Answer

Hey, everyone here we are asked to simplify the following expression and express the answer in standard form. If necessary. Here we are given the expression the quantity of negative six plus the square root of negative 64 squared. Here we have four answer choice options, answer choice, A negative 28 minus 96. I answer B negative 28 plus 96. I answer C negative 12 plus 16 I and answer D minus 16. I. So looking at our given expression, we can notice that this expression follows the formula where we have the quantity of X plus Y squared is equal to X squared plus two XY plus Y squared. So now we just want to compare our expression to our formula to identify our values for X and Y. So starting with our value for XX is going to be the first term here which is negative six. And then our second term is going to be four Y. So Y is the square root of negative 64. And now we can simplify the square root of negative 64 by breaking up the value underneath the root. So breaking up the value, we have the square root of 64 multiplied by negative one. And now, if we recall the square root of 64 is eight, so we now have eight multiplied by neg the square root of negative one. And now we just need to recall that the square root of negative one is equivalent to I. So rewriting with I, we see that Y simplifies to eight I. So now we can start filling in our formula here. So we have the quantity of negative six plus the square root of negative 64 which we turn determined simplifies to A I. So we have negative six plus A I and this whole quantity is squared and this is equal to X squared, which we now have as the quantity of negative six squared, we then have plus two multiplied by negative six multiplied by A I. Then we have plus Y squared. So we have plus the quantity of A I squared. And now we just want to start simplifying. So our first term again, we have negative six quantity squared which gives us 36 we then have plus two times negative six, which gives us negative 12. So we have minus 12 multiplied by A I. Then we have plus the quantity of eight I squared which gives us plus 64 I squared. And so now we need to continue to simplify. So we have 36 and now 12 times eight gives us 96. So we have again 36 minus 96 I, then we need to recall that I is the square root of negative one. So when we have I squared, we have the quantity of the square root of negative one squared. And this just simplifies to negative one. So I squared gives us negative one. So we have plus 64 multiplied by negative one. And so now just simplifying one last time, we are left with 36 minus I minus 64. Now we just need to combine like terms and we are left with negative 28 minus 96 I and with our final simplified expression, we are left with answer choice. A where again we have negative 28 minus 96. I thanks for watching. I hope you found this video helpful and I'll see you again next time.

In Exercises 1–10, perform the indicated operations and write the result in standard form. ___ (−2 + √−100)²

Key Concepts

Complex Numbers

Squaring a Binomial

Standard Form of Complex Numbers

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