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Popular >

simplify (4-3i)^5

Solution

simplify

(4 - 3 i)^{5}

Solution

- 3116 + 237 i

Solution steps

(4 - 3 i)^{5}

Apply binomial theorem:

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

a = 4, b = - 3 i

= i = 0 \sum 5 (i 5) \cdot 4^{(5 - i)} (- 3 i)^{i}

Expand summation

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

Simplify

\frac{5 !}{0 ! ( 5 - 0 ) !} \cdot 4^{5} (- 3 i)^{0} : 1024

Simplify

\frac{5 !}{1 ! ( 5 - 1 ) !} \cdot 4^{4} (- 3 i)^{1} : - 3840 i

Simplify

\frac{5 !}{2 ! ( 5 - 2 ) !} \cdot 4^{3} (- 3 i)^{2} : 5760 i^{2}

Simplify

\frac{5 !}{3 ! ( 5 - 3 ) !} \cdot 4^{2} (- 3 i)^{3} : - 4320 i^{3}

Simplify

\frac{5 !}{4 ! ( 5 - 4 ) !} \cdot 4^{1} (- 3 i)^{4} : 1620 i^{4}

Simplify

\frac{5 !}{5 ! ( 5 - 5 ) !} \cdot 4^{0} (- 3 i)^{5} : - 243 i^{5}

= 1024 - 3840 i + 5760 i^{2} - 4320 i^{3} + 1620 i^{4} - 243 i^{5}

Simplify

1024 - 3840 i + 5760 i^{2} - 4320 i^{3} + 1620 i^{4} - 243 i^{5} : 237 i - 3116

= 237 i - 3116

Rewrite in standard complex form:

- 3116 + 237 i

= - 3116 + 237 i

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Frequently Asked Questions (FAQ)

What is (4-3i)^5 ?
The solution to (4-3i)^5 is -3116+237i