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Popular Algebra >

simplify (3+3i)^7

Solution

simplify

(3 + 3 i)^{7}

Solution

17496 - 17496 i

Solution steps

(3 + 3 i)^{7}

Apply binomial theorem:

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

a = 3, b = 3 i

= i = 0 \sum 7 (i 7) \cdot 3^{(7 - i)} (3 i)^{i}

Expand summation

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

Simplify

\frac{7 !}{0 ! ( 7 - 0 ) !} \cdot 3^{7} (3 i)^{0} : 2187

Simplify

\frac{7 !}{1 ! ( 7 - 1 ) !} \cdot 3^{6} (3 i)^{1} : 15309 i

Simplify

\frac{7 !}{2 ! ( 7 - 2 ) !} \cdot 3^{5} (3 i)^{2} : 45927 i^{2}

Simplify

\frac{7 !}{3 ! ( 7 - 3 ) !} \cdot 3^{4} (3 i)^{3} : 76545 i^{3}

Simplify

\frac{7 !}{4 ! ( 7 - 4 ) !} \cdot 3^{3} (3 i)^{4} : 76545 i^{4}

Simplify

\frac{7 !}{5 ! ( 7 - 5 ) !} \cdot 3^{2} (3 i)^{5} : 45927 i^{5}

Simplify

\frac{7 !}{6 ! ( 7 - 6 ) !} \cdot 3^{1} (3 i)^{6} : 15309 i^{6}

Simplify

\frac{7 !}{7 ! ( 7 - 7 ) !} \cdot 3^{0} (3 i)^{7} : 2187 i^{7}

= 2187 + 15309 i + 45927 i^{2} + 76545 i^{3} + 76545 i^{4} + 45927 i^{5} + 15309 i^{6} + 2187 i^{7}

Simplify

2187 + 15309 i + 45927 i^{2} + 76545 i^{3} + 76545 i^{4} + 45927 i^{5} + 15309 i^{6} + 2187 i^{7} : - 17496 i + 17496

= - 17496 i + 17496

Rewrite in standard complex form:

17496 - 17496 i

= 17496 - 17496 i

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Frequently Asked Questions (FAQ)

What is (3+3i)^7 ?
The solution to (3+3i)^7 is 17496-17496i

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