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Popular >

simplify (5+5i)^5

Solution

simplify

(5 + 5 i)^{5}

Solution

- 12500 - 12500 i

Solution steps

(5 + 5 i)^{5}

Apply binomial theorem:

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

a = 5, b = 5 i

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

Expand summation

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

Simplify

\frac{5 !}{0 ! ( 5 - 0 ) !} \cdot 5^{5} (5 i)^{0} : 3125

Simplify

\frac{5 !}{1 ! ( 5 - 1 ) !} \cdot 5^{4} (5 i)^{1} : 15625 i

Simplify

\frac{5 !}{2 ! ( 5 - 2 ) !} \cdot 5^{3} (5 i)^{2} : 31250 i^{2}

Simplify

\frac{5 !}{3 ! ( 5 - 3 ) !} \cdot 5^{2} (5 i)^{3} : 31250 i^{3}

Simplify

\frac{5 !}{4 ! ( 5 - 4 ) !} \cdot 5^{1} (5 i)^{4} : 15625 i^{4}

Simplify

\frac{5 !}{5 ! ( 5 - 5 ) !} \cdot 5^{0} (5 i)^{5} : 3125 i^{5}

= 3125 + 15625 i + 31250 i^{2} + 31250 i^{3} + 15625 i^{4} + 3125 i^{5}

Simplify

3125 + 15625 i + 31250 i^{2} + 31250 i^{3} + 15625 i^{4} + 3125 i^{5} : - 12500 i - 12500

= - 12500 i - 12500

Rewrite in standard complex form:

- 12500 - 12500 i

= - 12500 - 12500 i

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Frequently Asked Questions (FAQ)

What is (5+5i)^5 ?
The solution to (5+5i)^5 is -12500-12500i