(2-2i)^5
find the given power write answer in rectangular form
The answer is -128+128i
To find the fifth power of (2 - 2i) and express the answer in rectangular form, we can use the binomial theorem or the rules of exponents.
Using the binomial theorem:
(2 - 2i)^5 = C(5,0)(2)^5(-2i)^0 + C(5,1)(2)^4(-2i)^1 + C(5,2)(2)^3(-2i)^2 + C(5,3)(2)^2(-2i)^3 + C(5,4)(2)^1(-2i)^4 + C(5,5)(2)^0(-2i)^5
Simplifying this expression:
(2 - 2i)^5 = 1(32)(1) + 5(16)(-2i) + 10(8)(4) + 10(4)(-8i) + 5(2)(16) + 1(1)(-32i)
(2 - 2i)^5 = 32 - 160i + 320 - 320i + 160 - 32i
(2 - 2i)^5 = 512 - 512i
Therefore, the fifth power of (2 - 2i) in rectangular form is 512 - 512i.
To find the value of the expression (2 - 2i)^5 and write the answer in rectangular form, we can use De Moivre's theorem. De Moivre's theorem states that for any complex number z = r(cosθ + isinθ), its nth power can be found by raising r to the power of n and multiplying the result by (cos(nθ) + isin(nθ)).
Let's break down the given expression and convert it into polar form:
2 - 2i = 2(cos(-π/4) + isin(-π/4))
Now, we can use De Moivre's theorem to find its fifth power:
(2 - 2i)^5 = (2(cos(-π/4) + isin(-π/4)))^5
Using the power rule, we can raise the magnitude to the fifth power and multiply the angle by 5:
= 2^5 (cos(-π/4 * 5) + isin(-π/4 * 5))
Simplifying further:
= 32(cos(-5π/4) + isin(-5π/4))
Now, we can convert the result back to rectangular form by expanding the polar form using Euler's formula:
= 32(cos(-5π/4) + isin(-5π/4))
= 32(-√2/2 - i√2/2) [Using cos(-5π/4) = -√2/2 and sin(-5π/4) = -√2/2]
Expanding further:
= 32(-√2/2 - i√2/2)
= -16√2 - 16√2i
Therefore, the expression (2 - 2i)^5 in rectangular form is -16√2 - 16√2i.