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Popular >

simplify (2+2i)^8

Solution

simplify

(2 + 2 i)^{8}

Solution

4096

Solution steps

(2 + 2 i)^{8}

Apply binomial theorem:

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

a = 2, b = 2 i

= i = 0 \sum 8 (i 8) \cdot 2^{(8 - i)} (2 i)^{i}

Expand summation

= \frac{8 !}{0 ! ( 8 - 0 ) !} \cdot 2^{8} (2 i)^{0} + \frac{8 !}{1 ! ( 8 - 1 ) !} \cdot 2^{7} (2 i)^{1} + \frac{8 !}{2 ! ( 8 - 2 ) !} \cdot 2^{6} (2 i)^{2} + \frac{8 !}{3 ! ( 8 - 3 ) !} \cdot 2^{5} (2 i)^{3} + \frac{8 !}{4 ! ( 8 - 4 ) !} \cdot 2^{4} (2 i)^{4} + \frac{8 !}{5 ! ( 8 - 5 ) !} \cdot 2^{3} (2 i)^{5} + \frac{8 !}{6 ! ( 8 - 6 ) !} \cdot 2^{2} (2 i)^{6} + \frac{8 !}{7 ! ( 8 - 7 ) !} \cdot 2^{1} (2 i)^{7} + \frac{8 !}{8 ! ( 8 - 8 ) !} \cdot 2^{0} (2 i)^{8}

Simplify

\frac{8 !}{0 ! ( 8 - 0 ) !} \cdot 2^{8} (2 i)^{0} : 256

Simplify

\frac{8 !}{1 ! ( 8 - 1 ) !} \cdot 2^{7} (2 i)^{1} : 2048 i

Simplify

\frac{8 !}{2 ! ( 8 - 2 ) !} \cdot 2^{6} (2 i)^{2} : 7168 i^{2}

Simplify

\frac{8 !}{3 ! ( 8 - 3 ) !} \cdot 2^{5} (2 i)^{3} : 14336 i^{3}

Simplify

\frac{8 !}{4 ! ( 8 - 4 ) !} \cdot 2^{4} (2 i)^{4} : 17920 i^{4}

Simplify

\frac{8 !}{5 ! ( 8 - 5 ) !} \cdot 2^{3} (2 i)^{5} : 14336 i^{5}

Simplify

\frac{8 !}{6 ! ( 8 - 6 ) !} \cdot 2^{2} (2 i)^{6} : 7168 i^{6}

Simplify

\frac{8 !}{7 ! ( 8 - 7 ) !} \cdot 2^{1} (2 i)^{7} : 2048 i^{7}

Simplify

\frac{8 !}{8 ! ( 8 - 8 ) !} \cdot 2^{0} (2 i)^{8} : 256 i^{8}

= 256 + 2048 i + 7168 i^{2} + 14336 i^{3} + 17920 i^{4} + 14336 i^{5} + 7168 i^{6} + 2048 i^{7} + 256 i^{8}

Simplify

256 + 2048 i + 7168 i^{2} + 14336 i^{3} + 17920 i^{4} + 14336 i^{5} + 7168 i^{6} + 2048 i^{7} + 256 i^{8} : 4096

= 4096

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Frequently Asked Questions (FAQ)

What is (2+2i)^8 ?
The solution to (2+2i)^8 is 4096