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Popular >

simplify (1-3i)^5

Solution

simplify

(1 - 3 i)^{5}

Solution

316 + 12 i

Solution steps

(1 - 3 i)^{5}

Apply binomial theorem:

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

a = 1, b = - 3 i

= i = 0 \sum 5 (i 5) \cdot 1^{(5 - i)} (- 3 i)^{i}

Expand summation

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

Simplify

\frac{5 !}{0 ! ( 5 - 0 ) !} \cdot 1^{5} (- 3 i)^{0} : 1

Simplify

\frac{5 !}{1 ! ( 5 - 1 ) !} \cdot 1^{4} (- 3 i)^{1} : - 15 i

Simplify

\frac{5 !}{2 ! ( 5 - 2 ) !} \cdot 1^{3} (- 3 i)^{2} : 90 i^{2}

Simplify

\frac{5 !}{3 ! ( 5 - 3 ) !} \cdot 1^{2} (- 3 i)^{3} : - 270 i^{3}

Simplify

\frac{5 !}{4 ! ( 5 - 4 ) !} \cdot 1^{1} (- 3 i)^{4} : 405 i^{4}

Simplify

\frac{5 !}{5 ! ( 5 - 5 ) !} \cdot 1^{0} (- 3 i)^{5} : - 243 i^{5}

= 1 - 15 i + 90 i^{2} - 270 i^{3} + 405 i^{4} - 243 i^{5}

Simplify

1 - 15 i + 90 i^{2} - 270 i^{3} + 405 i^{4} - 243 i^{5} : 12 i + 316

= 12 i + 316

Rewrite in standard complex form:

316 + 12 i

= 316 + 12 i

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Frequently Asked Questions (FAQ)

What is (1-3i)^5 ?
The solution to (1-3i)^5 is 316+12i