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Popular >

simplify (3+4i)^6

Solution

simplify

(3 + 4 i)^{6}

Solution

11753 - 10296 i

Solution steps

(3 + 4 i)^{6}

Apply binomial theorem:

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

a = 3, b = 4 i

= i = 0 \sum 6 (i 6) \cdot 3^{(6 - i)} (4 i)^{i}

Expand summation

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

Simplify

\frac{6 !}{0 ! ( 6 - 0 ) !} \cdot 3^{6} (4 i)^{0} : 729

Simplify

\frac{6 !}{1 ! ( 6 - 1 ) !} \cdot 3^{5} (4 i)^{1} : 5832 i

Simplify

\frac{6 !}{2 ! ( 6 - 2 ) !} \cdot 3^{4} (4 i)^{2} : 19440 i^{2}

Simplify

\frac{6 !}{3 ! ( 6 - 3 ) !} \cdot 3^{3} (4 i)^{3} : 34560 i^{3}

Simplify

\frac{6 !}{4 ! ( 6 - 4 ) !} \cdot 3^{2} (4 i)^{4} : 34560 i^{4}

Simplify

\frac{6 !}{5 ! ( 6 - 5 ) !} \cdot 3^{1} (4 i)^{5} : 18432 i^{5}

Simplify

\frac{6 !}{6 ! ( 6 - 6 ) !} \cdot 3^{0} (4 i)^{6} : 4096 i^{6}

= 729 + 5832 i + 19440 i^{2} + 34560 i^{3} + 34560 i^{4} + 18432 i^{5} + 4096 i^{6}

Simplify

729 + 5832 i + 19440 i^{2} + 34560 i^{3} + 34560 i^{4} + 18432 i^{5} + 4096 i^{6} : - 10296 i + 11753

= - 10296 i + 11753

Rewrite in standard complex form:

11753 - 10296 i

= 11753 - 10296 i

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Frequently Asked Questions (FAQ)

What is (3+4i)^6 ?
The solution to (3+4i)^6 is 11753-10296i