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Popular Álgebra >

simplificar (2+i)^6

Solução

simplificar

(2 + i)^{6}

Solução

- 117 + 44 i

Passos da solução

(2 + i)^{6}

Aplicar o teorema do binômio:

(a + b)^{n} = i = 0 \sum n (i n) a^{(n - i)} b^{i}

a = 2, b = i

= i = 0 \sum 6 (i 6) \cdot 2^{(6 - i)} i^{i}

Expandir a soma

= \frac{6 !}{0 ! ( 6 - 0 ) !} \cdot 2^{6} i^{0} + \frac{6 !}{1 ! ( 6 - 1 ) !} \cdot 2^{5} i^{1} + \frac{6 !}{2 ! ( 6 - 2 ) !} \cdot 2^{4} i^{2} + \frac{6 !}{3 ! ( 6 - 3 ) !} \cdot 2^{3} i^{3} + \frac{6 !}{4 ! ( 6 - 4 ) !} \cdot 2^{2} i^{4} + \frac{6 !}{5 ! ( 6 - 5 ) !} \cdot 2^{1} i^{5} + \frac{6 !}{6 ! ( 6 - 6 ) !} \cdot 2^{0} i^{6}

Simplificar

\frac{6 !}{0 ! ( 6 - 0 ) !} \cdot 2^{6} i^{0} : 64

Simplificar

\frac{6 !}{1 ! ( 6 - 1 ) !} \cdot 2^{5} i^{1} : 192 i

Simplificar

\frac{6 !}{2 ! ( 6 - 2 ) !} \cdot 2^{4} i^{2} : 240 i^{2}

Simplificar

\frac{6 !}{3 ! ( 6 - 3 ) !} \cdot 2^{3} i^{3} : 160 i^{3}

Simplificar

\frac{6 !}{4 ! ( 6 - 4 ) !} \cdot 2^{2} i^{4} : 60 i^{4}

Simplificar

\frac{6 !}{5 ! ( 6 - 5 ) !} \cdot 2^{1} i^{5} : 12 i^{5}

Simplificar

\frac{6 !}{6 ! ( 6 - 6 ) !} \cdot 2^{0} i^{6} : i^{6}

= 64 + 192 i + 240 i^{2} + 160 i^{3} + 60 i^{4} + 12 i^{5} + i^{6}

Simplificar

64 + 192 i + 240 i^{2} + 160 i^{3} + 60 i^{4} + 12 i^{5} + i^{6} : 44 i - 117

= 44 i - 117

Rewrite in standard complex form:

- 117 + 44 i

= - 117 + 44 i

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