# ind The quantity of 1 + i all to the fifth power. and write the answer in standard form.

Find  and write the answer in standard form.
 A. B. C. D.
When a complex number is in trigonometric form , what does r represent?
 A. abscissa B. argument C. imaginary unit D. modulus

(b)

The complex number x+iy  corresponds to the point with coordinates (x, y) .

x is a real part of complex number .

y is a imaginary part of complex number .

The polar form of  x+iy is r(cos ø +i sin ø ) .

Where

r is  called the modulus - is the absolute value of the hypotenuse formed by sides "x" and "y" .

r = (x2 + y2)1/2

θ is called the argument .

θ = arc tan (y/x)

Hence r is called modulus .

So option (D) is correct .

selected Oct 21, 2014 by tonymate

The complex number is   .

Convert the complex number 1+i into polor form .

r = (x2 + y2)1/2        θ = arc tan (y/x)

r = (12 + 12)1/2        θ = arc tan (1/1)

r = (2)1/2                   θ = arc tan (1/1)

r = √2                    θ = arc tan (1)

r = √2                    θ = π/4

The polar form of  x+iy is r(cos ø +i sin ø ) .

1+ i = √2  (cos π/4 +i sin π/4 )

De Movire's Theorem states that:

(cos x + i sin x)^n = cos(nx) + i sin(nx)

So  (1 + i)^5 = (√2)^5[cos(5 * π/4) + i sin (5 * π/4)]

(1 + i)^5 = (4√2)[cos( 5π/4) + i sin (5π/4)]

(1 + i)^5 = (4√2)[-1/√2 -i/√2]

(1 + i)^5 = (4√2)(1/√2 )[-1-i]

(1 + i)^5 = 4[-1-i]

=  -4 - 4i

Hence option (B) is correct .