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Popular >

simplify (4+4i)^5

Solution

simplify

(4 + 4 i)^{5}

Solution

- 4096 - 4096 i

Solution steps

(4 + 4 i)^{5}

Apply binomial theorem:

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

a = 4, b = 4 i

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

Expand summation

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

Simplify

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

Simplify

\frac{5 !}{1 ! ( 5 - 1 ) !} \cdot 4^{4} (4 i)^{1} : 5120 i

Simplify

\frac{5 !}{2 ! ( 5 - 2 ) !} \cdot 4^{3} (4 i)^{2} : 10240 i^{2}

Simplify

\frac{5 !}{3 ! ( 5 - 3 ) !} \cdot 4^{2} (4 i)^{3} : 10240 i^{3}

Simplify

\frac{5 !}{4 ! ( 5 - 4 ) !} \cdot 4^{1} (4 i)^{4} : 5120 i^{4}

Simplify

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

= 1024 + 5120 i + 10240 i^{2} + 10240 i^{3} + 5120 i^{4} + 1024 i^{5}

Simplify

1024 + 5120 i + 10240 i^{2} + 10240 i^{3} + 5120 i^{4} + 1024 i^{5} : - 4096 i - 4096

= - 4096 i - 4096

Rewrite in standard complex form:

- 4096 - 4096 i

= - 4096 - 4096 i

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Frequently Asked Questions (FAQ)

What is (4+4i)^5 ?
The solution to (4+4i)^5 is -4096-4096i