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

Question

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

 

Answer 1

(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 .

Answer 2

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 .