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Popular >

simplify (3+2i)^6

Solution

simplify

(3 + 2 i)^{6}

Solution

- 2035 - 828 i

Solution steps

(3 + 2 i)^{6}

Apply binomial theorem:

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

a = 3, b = 2 i

= i = 0 \sum 6 (i 6) \cdot 3^{(6 - i)} (2 i)^{i}

Expand summation

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

Simplify

\frac{6 !}{0 ! ( 6 - 0 ) !} \cdot 3^{6} (2 i)^{0} : 729

Simplify

\frac{6 !}{1 ! ( 6 - 1 ) !} \cdot 3^{5} (2 i)^{1} : 2916 i

Simplify

\frac{6 !}{2 ! ( 6 - 2 ) !} \cdot 3^{4} (2 i)^{2} : 4860 i^{2}

Simplify

\frac{6 !}{3 ! ( 6 - 3 ) !} \cdot 3^{3} (2 i)^{3} : 4320 i^{3}

Simplify

\frac{6 !}{4 ! ( 6 - 4 ) !} \cdot 3^{2} (2 i)^{4} : 2160 i^{4}

Simplify

\frac{6 !}{5 ! ( 6 - 5 ) !} \cdot 3^{1} (2 i)^{5} : 576 i^{5}

Simplify

\frac{6 !}{6 ! ( 6 - 6 ) !} \cdot 3^{0} (2 i)^{6} : 64 i^{6}

= 729 + 2916 i + 4860 i^{2} + 4320 i^{3} + 2160 i^{4} + 576 i^{5} + 64 i^{6}

Simplify

729 + 2916 i + 4860 i^{2} + 4320 i^{3} + 2160 i^{4} + 576 i^{5} + 64 i^{6} : - 828 i - 2035

= - 828 i - 2035

Rewrite in standard complex form:

- 2035 - 828 i

= - 2035 - 828 i

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Frequently Asked Questions (FAQ)

What is (3+2i)^6 ?
The solution to (3+2i)^6 is -2035-828i