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Popular Álgebra >

simplificar (4+9i)^6

Solução

simplificar

(4 + 9 i)^{6}

Solução

736255 + 539352 i

Passos da solução

(4 + 9 i)^{6}

Aplicar o teorema do binômio:

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

a = 4, b = 9 i

= i = 0 \sum 6 (i 6) \cdot 4^{(6 - i)} (9 i)^{i}

Expandir a soma

= \frac{6 !}{0 ! ( 6 - 0 ) !} \cdot 4^{6} (9 i)^{0} + \frac{6 !}{1 ! ( 6 - 1 ) !} \cdot 4^{5} (9 i)^{1} + \frac{6 !}{2 ! ( 6 - 2 ) !} \cdot 4^{4} (9 i)^{2} + \frac{6 !}{3 ! ( 6 - 3 ) !} \cdot 4^{3} (9 i)^{3} + \frac{6 !}{4 ! ( 6 - 4 ) !} \cdot 4^{2} (9 i)^{4} + \frac{6 !}{5 ! ( 6 - 5 ) !} \cdot 4^{1} (9 i)^{5} + \frac{6 !}{6 ! ( 6 - 6 ) !} \cdot 4^{0} (9 i)^{6}

Simplificar

\frac{6 !}{0 ! ( 6 - 0 ) !} \cdot 4^{6} (9 i)^{0} : 4096

Simplificar

\frac{6 !}{1 ! ( 6 - 1 ) !} \cdot 4^{5} (9 i)^{1} : 55296 i

Simplificar

\frac{6 !}{2 ! ( 6 - 2 ) !} \cdot 4^{4} (9 i)^{2} : 311040 i^{2}

Simplificar

\frac{6 !}{3 ! ( 6 - 3 ) !} \cdot 4^{3} (9 i)^{3} : 933120 i^{3}

Simplificar

\frac{6 !}{4 ! ( 6 - 4 ) !} \cdot 4^{2} (9 i)^{4} : 1574640 i^{4}

Simplificar

\frac{6 !}{5 ! ( 6 - 5 ) !} \cdot 4^{1} (9 i)^{5} : 1417176 i^{5}

Simplificar

\frac{6 !}{6 ! ( 6 - 6 ) !} \cdot 4^{0} (9 i)^{6} : 531441 i^{6}

= 4096 + 55296 i + 311040 i^{2} + 933120 i^{3} + 1574640 i^{4} + 1417176 i^{5} + 531441 i^{6}

Simplificar

4096 + 55296 i + 311040 i^{2} + 933120 i^{3} + 1574640 i^{4} + 1417176 i^{5} + 531441 i^{6} : 539352 i + 736255

= 539352 i + 736255

Rewrite in standard complex form:

736255 + 539352 i

= 736255 + 539352 i

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