What is expand (3+5i)^4 ?

The solution to expand (3+5i)^4 is -644-960i

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expand (3+5i)^4

expand

(3 + 5 i)^{4}

- 644 - 960 i

Solution steps

(3 + 5 i)^{4}

Apply binomial theorem:

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

a = 3, b = 5 i

= i = 0 \sum 4 (i 4) \cdot 3^{(4 - i)} (5 i)^{i}

Expand summation

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

Simplify

\frac{4 !}{0 ! ( 4 - 0 ) !} \cdot 3^{4} (5 i)^{0} : 81

Simplify

\frac{4 !}{1 ! ( 4 - 1 ) !} \cdot 3^{3} (5 i)^{1} : 540 i

Simplify

\frac{4 !}{2 ! ( 4 - 2 ) !} \cdot 3^{2} (5 i)^{2} : 1350 i^{2}

Simplify

\frac{4 !}{3 ! ( 4 - 3 ) !} \cdot 3^{1} (5 i)^{3} : 1500 i^{3}

Simplify

\frac{4 !}{4 ! ( 4 - 4 ) !} \cdot 3^{0} (5 i)^{4} : 625 i^{4}

= 81 + 540 i + 1350 i^{2} + 1500 i^{3} + 625 i^{4}

Simplify

81 + 540 i + 1350 i^{2} + 1500 i^{3} + 625 i^{4} : - 960 i - 644

= - 960 i - 644

Rewrite in standard complex form:

- 644 - 960 i

= - 644 - 960 i

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