In Exercises 53–64, use DeMoivre's Theorem to find the indicat...

Question

In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. [√3 (cos (5π/18) + i sin (5π/18))]⁶

Accepted Answer

Welcome back. Everyone. In this problem, we want to find the square root of seven multiplied by the cosine of 2/9 of pi plus I multiplied by the sign of 2/9 of pi all to the core of six using the theorem. And we want to express our answer in a rectangular form. That is the standard form for complex numbers. For our answer choices. A is 343 plus 343. I multiplied by the skirt of three divided by two B is negative 343 divided by two minus 343. I multiplied by the skirt of three divided by two C equals negative 343 multiplied by the skirt of two, divided by two minus 343 divided by two I and D is negative 343 multiplied by the skirt of three divided by two plus 343. I multiplied by the square root of two divided by two. Now, what do we already know? Well, we know that right here we have a complex number. OK. And we want to find the answer to that complex number. Being raised to the poor using the theorem. What do we know about that? Well, recall that the marvelous theorem tells us that if we have a complex number that's written in the form R multiplied by the cosine of theta plus I multiplied by the sign of theta, which is what we have here in our complex number. Then if we raise that complex number to a power, we raise the constant. It's being multiplied by, to a power that is R and we multiply of our angles or values of the, by the value of the power. So that means for our number that we have here, that's exactly what we will need to do. First, let's see if we can identify what those values are. So R is our constant. In this case, notice that R equals seven. Additionally, our power is, well, the number is being raised to the power of six. So that means N equals six. So we can substitute these values into what we know about the Marous theorem to solve for or complex number being raised to that power. So that tells us that raising our complex number to the sixth part means we're going to raise the square root of seven to the sixth power. And we're going to multiply our angles in this case, 2/9 of pi by the value of the poor, which is six. So that would be the core sign off six multiplied by 2/9. Of pi plus I multiplied by the sign off six multiplied by 2/9 of pi no. Where do we go from here? Well, if we're raising the square of seven to the poor of six, recall that we can rewrite the square root as a fractional power. In other words, the spirit of seven can be rewritten as seven to the poor of a half. So if we are raising it to the sixth power, when we raise one power to another, we can multiply them. So this can be rewritten as seven to the power of half multiplied by 6.5 of six is three. So that means I can replace the skirt of seven wrist to the power of six with seven cubed. I should probably rewrite these in blue to keep us on the same track. Next, we're finding the cosine of six multiplied by 2/9 of pi. Well, we can factor here because three factored from six is two and three factored from nine is three. And when we go ahead and multiply two multiplied by two pi, that's going to be four pi. So we can simplify that as the core sign of four thirds of pi. Likewise, the same will happen with I multiplied by a sign. We can rewrite that as four thirds of pi. No, we're almost there. We can rewrite several thirds as 343. OK? And no, we just need to figure out what are the values of the core sign of four thirds of pi and the sign of four thirds of pi. Well, by our unit circle recall that the co sign of four thirds of pie equals negative or half and the sign of four thirds of pi, OK equals the negative square root of three divided by two. So now we can substitute our values into our equation. And now that becomes 343 multiplied by the cosine of four thirds of pi which is negative a half plus I sign of four thirds of pie. I got all of that. So that's gonna be I times the negative skirt of three divided by two, no 343 multiplied by negative A half. That's negative 343 divided by two and 343 multiplied by the negative square of three divided by two. I will be negative multi I multiplied by the square of three are divided by two. So our answer is going to be answer choice. B negative 343 divided by two minus 343. I multiplied by the square of three divided by two. Thanks for watching everybody. I hope this video helped.

In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. [√3 (cos (5π/18) + i sin (5π/18))]⁶

Key Concepts

DeMoivre's Theorem

Polar and Rectangular Forms of Complex Numbers

Trigonometric Identities and Angle Multiplication

Watch next