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Popular Algebra >

simplify (4+5i)^7

Solution

simplify

(4 + 5 i)^{7}

Solution

441284 - 4765 i

Solution steps

(4 + 5 i)^{7}

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 7 (i 7) \cdot 4^{(7 - i)} (5 i)^{i}

Expand summation

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

Simplify

\frac{7 !}{0 ! ( 7 - 0 ) !} \cdot 4^{7} (5 i)^{0} : 16384

Simplify

\frac{7 !}{1 ! ( 7 - 1 ) !} \cdot 4^{6} (5 i)^{1} : 143360 i

Simplify

\frac{7 !}{2 ! ( 7 - 2 ) !} \cdot 4^{5} (5 i)^{2} : 537600 i^{2}

Simplify

\frac{7 !}{3 ! ( 7 - 3 ) !} \cdot 4^{4} (5 i)^{3} : 1120000 i^{3}

Simplify

\frac{7 !}{4 ! ( 7 - 4 ) !} \cdot 4^{3} (5 i)^{4} : 1400000 i^{4}

Simplify

\frac{7 !}{5 ! ( 7 - 5 ) !} \cdot 4^{2} (5 i)^{5} : 1050000 i^{5}

Simplify

\frac{7 !}{6 ! ( 7 - 6 ) !} \cdot 4^{1} (5 i)^{6} : 437500 i^{6}

Simplify

\frac{7 !}{7 ! ( 7 - 7 ) !} \cdot 4^{0} (5 i)^{7} : 78125 i^{7}

= 16384 + 143360 i + 537600 i^{2} + 1120000 i^{3} + 1400000 i^{4} + 1050000 i^{5} + 437500 i^{6} + 78125 i^{7}

Simplify

16384 + 143360 i + 537600 i^{2} + 1120000 i^{3} + 1400000 i^{4} + 1050000 i^{5} + 437500 i^{6} + 78125 i^{7} : - 4765 i + 441284

= - 4765 i + 441284

Rewrite in standard complex form:

441284 - 4765 i

= 441284 - 4765 i

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e x p an d \frac{( 3 ^{x} + 4 ^{x} ) ^{2}}{1 2 ^{x}}

Frequently Asked Questions (FAQ)

What is (4+5i)^7 ?
The solution to (4+5i)^7 is 441284-4765i

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