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Popular >

simplify (5+2i)^4

Solution

simplify

(5 + 2 i)^{4}

Solution

41 + 840 i

Solution steps

(5 + 2 i)^{4}

Apply binomial theorem:

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

a = 5, b = 2 i

= i = 0 \sum 4 (i 4) \cdot 5^{(4 - i)} (2 i)^{i}

Expand summation

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

Simplify

\frac{4 !}{0 ! ( 4 - 0 ) !} \cdot 5^{4} (2 i)^{0} : 625

Simplify

\frac{4 !}{1 ! ( 4 - 1 ) !} \cdot 5^{3} (2 i)^{1} : 1000 i

Simplify

\frac{4 !}{2 ! ( 4 - 2 ) !} \cdot 5^{2} (2 i)^{2} : 600 i^{2}

Simplify

\frac{4 !}{3 ! ( 4 - 3 ) !} \cdot 5^{1} (2 i)^{3} : 160 i^{3}

Simplify

\frac{4 !}{4 ! ( 4 - 4 ) !} \cdot 5^{0} (2 i)^{4} : 16 i^{4}

= 625 + 1000 i + 600 i^{2} + 160 i^{3} + 16 i^{4}

Simplify

625 + 1000 i + 600 i^{2} + 160 i^{3} + 16 i^{4} : 840 i + 41

= 840 i + 41

Rewrite in standard complex form:

41 + 840 i

= 41 + 840 i

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Frequently Asked Questions (FAQ)

What is (5+2i)^4 ?
The solution to (5+2i)^4 is 41+840i