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Popular >

simplify (5-3i)^6

Solution

simplify

(5 - 3 i)^{6}

Solution

- 39104 + 3960 i

Solution steps

(5 - 3 i)^{6}

Apply binomial theorem:

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

a = 5, b = - 3 i

= i = 0 \sum 6 (i 6) \cdot 5^{(6 - i)} (- 3 i)^{i}

Expand summation

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

Simplify

\frac{6 !}{0 ! ( 6 - 0 ) !} \cdot 5^{6} (- 3 i)^{0} : 15625

Simplify

\frac{6 !}{1 ! ( 6 - 1 ) !} \cdot 5^{5} (- 3 i)^{1} : - 56250 i

Simplify

\frac{6 !}{2 ! ( 6 - 2 ) !} \cdot 5^{4} (- 3 i)^{2} : 84375 i^{2}

Simplify

\frac{6 !}{3 ! ( 6 - 3 ) !} \cdot 5^{3} (- 3 i)^{3} : - 67500 i^{3}

Simplify

\frac{6 !}{4 ! ( 6 - 4 ) !} \cdot 5^{2} (- 3 i)^{4} : 30375 i^{4}

Simplify

\frac{6 !}{5 ! ( 6 - 5 ) !} \cdot 5^{1} (- 3 i)^{5} : - 7290 i^{5}

Simplify

\frac{6 !}{6 ! ( 6 - 6 ) !} \cdot 5^{0} (- 3 i)^{6} : 729 i^{6}

= 15625 - 56250 i + 84375 i^{2} - 67500 i^{3} + 30375 i^{4} - 7290 i^{5} + 729 i^{6}

Simplify

15625 - 56250 i + 84375 i^{2} - 67500 i^{3} + 30375 i^{4} - 7290 i^{5} + 729 i^{6} : 3960 i - 39104

= 3960 i - 39104

Rewrite in standard complex form:

- 39104 + 3960 i

= - 39104 + 3960 i

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Frequently Asked Questions (FAQ)

What is (5-3i)^6 ?
The solution to (5-3i)^6 is -39104+3960i