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Popular >

simplify (4+5i)^4

Solution

simplify

(4 + 5 i)^{4}

Solution

- 1519 - 720 i

Solution steps

(4 + 5 i)^{4}

Apply binomial theorem:

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

a = 4, b = 5 i

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

Expand summation

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

Simplify

\frac{4 !}{0 ! ( 4 - 0 ) !} \cdot 4^{4} (5 i)^{0} : 256

Simplify

\frac{4 !}{1 ! ( 4 - 1 ) !} \cdot 4^{3} (5 i)^{1} : 1280 i

Simplify

\frac{4 !}{2 ! ( 4 - 2 ) !} \cdot 4^{2} (5 i)^{2} : 2400 i^{2}

Simplify

\frac{4 !}{3 ! ( 4 - 3 ) !} \cdot 4^{1} (5 i)^{3} : 2000 i^{3}

Simplify

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

= 256 + 1280 i + 2400 i^{2} + 2000 i^{3} + 625 i^{4}

Simplify

256 + 1280 i + 2400 i^{2} + 2000 i^{3} + 625 i^{4} : - 720 i - 1519

= - 720 i - 1519

Rewrite in standard complex form:

- 1519 - 720 i

= - 1519 - 720 i

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Frequently Asked Questions (FAQ)

What is (4+5i)^4 ?
The solution to (4+5i)^4 is -1519-720i