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Popular >

simplify (2+2i)^7

Solution

simplify

(2 + 2 i)^{7}

Solution

1024 - 1024 i

Solution steps

(2 + 2 i)^{7}

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 7 (i 7) \cdot 2^{(7 - i)} (2 i)^{i}

Expand summation

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

Simplify

\frac{7 !}{0 ! ( 7 - 0 ) !} \cdot 2^{7} (2 i)^{0} : 128

Simplify

\frac{7 !}{1 ! ( 7 - 1 ) !} \cdot 2^{6} (2 i)^{1} : 896 i

Simplify

\frac{7 !}{2 ! ( 7 - 2 ) !} \cdot 2^{5} (2 i)^{2} : 2688 i^{2}

Simplify

\frac{7 !}{3 ! ( 7 - 3 ) !} \cdot 2^{4} (2 i)^{3} : 4480 i^{3}

Simplify

\frac{7 !}{4 ! ( 7 - 4 ) !} \cdot 2^{3} (2 i)^{4} : 4480 i^{4}

Simplify

\frac{7 !}{5 ! ( 7 - 5 ) !} \cdot 2^{2} (2 i)^{5} : 2688 i^{5}

Simplify

\frac{7 !}{6 ! ( 7 - 6 ) !} \cdot 2^{1} (2 i)^{6} : 896 i^{6}

Simplify

\frac{7 !}{7 ! ( 7 - 7 ) !} \cdot 2^{0} (2 i)^{7} : 128 i^{7}

= 128 + 896 i + 2688 i^{2} + 4480 i^{3} + 4480 i^{4} + 2688 i^{5} + 896 i^{6} + 128 i^{7}

Simplify

128 + 896 i + 2688 i^{2} + 4480 i^{3} + 4480 i^{4} + 2688 i^{5} + 896 i^{6} + 128 i^{7} : - 1024 i + 1024

= - 1024 i + 1024

Rewrite in standard complex form:

1024 - 1024 i

= 1024 - 1024 i

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Frequently Asked Questions (FAQ)

What is (2+2i)^7 ?
The solution to (2+2i)^7 is 1024-1024i