Given m 2 5i determine the value of m4 and give your answe

Given m = -2 + 5i determine the value of m^4 and give your answer in both trigonometric and standard forms for complex numbers. Find all complex solutions of x^6 + 64 = 0 Write your answers in trigonometric form using degrees.

Solution

10)m=-2+5i

r=((-2)²+5²)=29

m=29((-2_/29)+(5_/29)i)

m=29((-2_/29)+(5_/29)i)

=>m=29(cos(111.8^o)+sin(111.8^o)i)

=>m⁴=(29(cos(111.8^o)+sin(111.8^o)i))⁴

=>m⁴=29²(cos(111.8*4)^o+sin(111.8*4)^oi)

=>m⁴=841(cos(447.21)^o+sin(447.21)^oi)

=>m⁴=841(cos(87.21)^o+sin(87.21)^oi)

=>m⁴=841(0.04875+0.99881i)

=>m⁴=41 +840i

m⁴=41 +840i,m⁴=(841cos(87.21)^o+841sin(87.21)^oi)

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11)x⁶+64=0

x⁶=-64

x⁶=64(-1+0i)

x⁶=64eⁱ,x⁶=64eⁱ³,x⁶=64eⁱ⁵,x⁶=64eⁱ⁷,x⁶=64eⁱ⁹,x⁶=64eⁱ¹¹

x=64^1/6e^i/6,x=64^1/6e^i3/6,x=64^1/6e^i5/6,x=64^1/6e^i7/6,x=64^1/6e^i9/6,x=64^1/6e^i11/6

x=((2cos30^o)+(2sin30^o)i),x=((2cos90^o)+(2sin90^o)i),x=((2cos150^o)+(2sin150^o)i),x=((2cos210^o)+(2sin210^o)i),x=((2cos270^o)+(2sin270^o)i),x=((2cos330^o)+(2sin330^o)i)