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Popular >

simplify (3+4i)^5

Solution

simplify

(3 + 4 i)^{5}

Solution

- 237 - 3116 i

Solution steps

(3 + 4 i)^{5}

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

Expand summation

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

Simplify

\frac{5 !}{0 ! ( 5 - 0 ) !} \cdot 3^{5} (4 i)^{0} : 243

Simplify

\frac{5 !}{1 ! ( 5 - 1 ) !} \cdot 3^{4} (4 i)^{1} : 1620 i

Simplify

\frac{5 !}{2 ! ( 5 - 2 ) !} \cdot 3^{3} (4 i)^{2} : 4320 i^{2}

Simplify

\frac{5 !}{3 ! ( 5 - 3 ) !} \cdot 3^{2} (4 i)^{3} : 5760 i^{3}

Simplify

\frac{5 !}{4 ! ( 5 - 4 ) !} \cdot 3^{1} (4 i)^{4} : 3840 i^{4}

Simplify

\frac{5 !}{5 ! ( 5 - 5 ) !} \cdot 3^{0} (4 i)^{5} : 1024 i^{5}

= 243 + 1620 i + 4320 i^{2} + 5760 i^{3} + 3840 i^{4} + 1024 i^{5}

Simplify

243 + 1620 i + 4320 i^{2} + 5760 i^{3} + 3840 i^{4} + 1024 i^{5} : - 3116 i - 237

= - 3116 i - 237

Rewrite in standard complex form:

- 237 - 3116 i

= - 237 - 3116 i

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Frequently Asked Questions (FAQ)

What is (3+4i)^5 ?
The solution to (3+4i)^5 is -237-3116i