Find The quantity of 2 + 2 times i all to the seventh power. and write the answer in standard form.

Question

Find  and write the answer in standard form. 

	A.
	B.
	C.
	D.

Write  in abbreviated trigonometric form. 

	A.
	B.
	C.
	D.

 

Answer 1

(2).

The expression is (2+2i)⁷.

To apply De Moivre’s Theorem, we must first write the complex number in polar form. Since the magnitude of (2+2i) is √[(2)² + (2)²] = 2√2, we begin by writing

Now

The option B is correct.

Answer 2

(1).

The polar form of a complex number z = a + bi is z = r (cos θ + i sin θ), where r = | z | = √(a² + b²), a = r cos θ, and b = r sin θ, and θ = tan^{- 1}(b / a) for a > 0 or θ = tan^{- 1}(b / a) + π or θ = tan^{- 1}(b / a) + 180^o for a < 0.

The complex expression is (-1+i)¹⁰.

The complex number is z = - 1 + i.

So, first find the absolute value of r .

r = | z | = √(a² + b²) = √[ (-1)² + (1)² ] = √2.

Now find the argument θ.

Since a = - 1 < 0, use the formula θ = tan^{- 1}(b / a) + π.

θ = tan^{- 1}[ 1/(- 1) ] + π = - π/4 + π = 3π/4.

Therefore, the polar form of - 1 + i  is about √2[ cos(3π/4) + i sin(3π/4) ].

Now

The option A is correct.

 

Answer 3

(1).

The complex number is z = 1 - i.

So, first find the absolute value of r .

r = | z | = √(a² + b²) = √[ (1)² + (-1)² ] = √2.

Now find the argument θ.

Since a = 1 > 0 and b < 0 , use the formula θ = tan^{- 1}(b / a) + 360^o.

θ = tan^{- 1}[ (-1)/(1) ] + 360^o = - 45^o + 360^o = 315^o.

Therefore, the polar form of 1 - i  is about √2[ cos(315^o) + i sin(315^o) ].

The abbreviated trigonometric form is √2 cis(315^o).

The option C is correct.