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simplificar ((sqrt(34)e^{1.0304i})^4)/((2+6i)^3)

Solución

simplificar

\frac{( 34 e ^{1.0304 i} ) ^{4}}{( 2 + 6 i ) ^{3}}

Solución

4.25284 \dots + 1.67139 \dots i

Pasos de solución

\frac{( 34 e ^{1.0304 i} ) ^{4}}{( 2 + 6 i ) ^{3}}

(34 e^{1.0304 i})^{4} = (34 (cos (1.0304) + i sin (1.0304)))^{4}

= \frac{( 34 ( cos ( 1.0304 ) + i sin ( 1.0304 ) ) ) ^{4}}{( 2 + 6 i ) ^{3}}

(2 + 6 i)^{3} = - 208 - 144 i

= \frac{( 34 ( cos ( 1.0304 ) + i sin ( 1.0304 ) ) ) ^{4}}{( - 208 - 144 i )}

(34 (cos (1.0304) + sin (1.0304) i))^{4} = 3 4^{2} (0.51447 \dots + 0.85750 \dots i)^{4}

= \frac{3 4 ^{2} ( 0.51447 \dots + 0.85750 \dots i ) ^{4}}{( - 208 - 144 i )}

Quitar los parentesis:

(- a) = - a

= \frac{3 4 ^{2} ( 0.51447 \dots + 0.85750 \dots i ) ^{4}}{- 208 - 144 i}

Factorizar

- 208 - 144 i : - 16 (13 + 9 i)

= - \frac{3 4 ^{2} ( 0.51447 \dots + 0.85750 \dots i ) ^{4}}{16 ( 13 + 9 i )}

Factorizar

3 4^{2} : 2^{2} \cdot 1 7^{2}

Factorizar

16 : 2^{4}

= - \frac{2 ^{2} \cdot 1 7 ^{2} ( 0.51447 \dots + 0.85750 \dots i ) ^{4}}{2 ^{4} ( 13 + 9 i )}

Cancelar

- \frac{2 ^{2} \cdot 1 7 ^{2} ( 0.51447 \dots + 0.85750 \dots i ) ^{4}}{2 ^{4} ( 13 + 9 i )} : - \frac{1 7 ^{2} ( 0.51447 \dots + 0.85750 \dots i ) ^{4}}{2 ^{2} ( 13 + 9 i )}

= - \frac{1 7 ^{2} ( 0.51447 \dots + 0.85750 \dots i ) ^{4}}{2 ^{2} ( 13 + 9 i )}

1 7^{2} = 289

= - \frac{289 ( 0.51447 \dots + 0.85750 \dots i ) ^{4}}{2 ^{2} ( 13 + 9 i )}

2^{2} = 4

= - \frac{289 ( 0.51447 \dots + 0.85750 \dots i ) ^{4}}{4 ( 13 + 9 i )}

Convertir el elemento a una forma decimal

\frac{289}{4} = 72.25

= - 72.25 \cdot \frac{( 0.51447 \dots + 0.85750 \dots i ) ^{4}}{( 13 + 9 i )}

Quitar los parentesis:

(a) = a

= - 72.25 \cdot \frac{( 0.51447 \dots + 0.85750 \dots i ) ^{4}}{13 + 9 i}

Multiplicar fracciones:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= - \frac{( 0.51447 \dots + 0.85750 \dots i ) ^{4} \cdot 72.25}{13 + 9 i}

Racionalizar

- \frac{72.25 ( 0.51447 \dots + 0.85750 \dots i ) ^{4}}{13 + 9 i} : - \frac{- 1063.21127 \dots - 417.84855 \dots i}{250}

= - \frac{- 1063.21127 \dots - 417.84855 \dots i}{250}

Reescribir

- \frac{- 1063.21127 \dots - 417.84855 \dots i}{250}

en la forma binómica:

4.25284 \dots + 1.67139 \dots i

= 4.25284 \dots + 1.67139 \dots i

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