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Popular >

simplify (2-i)^{10}

Solution

simplify

(2 - i)^{10}

Solution

- 237 + 3116 i

Solution steps

(2 - i)^{10}

Apply binomial theorem:

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

a = 2, b = - i

= i = 0 \sum 10 (i 10) \cdot 2^{(10 - i)} (- i)^{i}

Expand summation

= \frac{10 !}{0 ! ( 10 - 0 ) !} \cdot 2^{10} (- i)^{0} + \frac{10 !}{1 ! ( 10 - 1 ) !} \cdot 2^{9} (- i)^{1} + \frac{10 !}{2 ! ( 10 - 2 ) !} \cdot 2^{8} (- i)^{2} + \frac{10 !}{3 ! ( 10 - 3 ) !} \cdot 2^{7} (- i)^{3} + \frac{10 !}{4 ! ( 10 - 4 ) !} \cdot 2^{6} (- i)^{4} + \frac{10 !}{5 ! ( 10 - 5 ) !} \cdot 2^{5} (- i)^{5} + \frac{10 !}{6 ! ( 10 - 6 ) !} \cdot 2^{4} (- i)^{6} + \frac{10 !}{7 ! ( 10 - 7 ) !} \cdot 2^{3} (- i)^{7} + \frac{10 !}{8 ! ( 10 - 8 ) !} \cdot 2^{2} (- i)^{8} + \frac{10 !}{9 ! ( 10 - 9 ) !} \cdot 2^{1} (- i)^{9} + \frac{10 !}{10 ! ( 10 - 10 ) !} \cdot 2^{0} (- i)^{10}

Simplify

\frac{10 !}{0 ! ( 10 - 0 ) !} \cdot 2^{10} (- i)^{0} : 1024

Simplify

\frac{10 !}{1 ! ( 10 - 1 ) !} \cdot 2^{9} (- i)^{1} : - 5120 i

Simplify

\frac{10 !}{2 ! ( 10 - 2 ) !} \cdot 2^{8} (- i)^{2} : 11520 i^{2}

Simplify

\frac{10 !}{3 ! ( 10 - 3 ) !} \cdot 2^{7} (- i)^{3} : - 15360 i^{3}

Simplify

\frac{10 !}{4 ! ( 10 - 4 ) !} \cdot 2^{6} (- i)^{4} : 13440 i^{4}

Simplify

\frac{10 !}{5 ! ( 10 - 5 ) !} \cdot 2^{5} (- i)^{5} : - 8064 i^{5}

Simplify

\frac{10 !}{6 ! ( 10 - 6 ) !} \cdot 2^{4} (- i)^{6} : 3360 i^{6}

Simplify

\frac{10 !}{7 ! ( 10 - 7 ) !} \cdot 2^{3} (- i)^{7} : - 960 i^{7}

Simplify

\frac{10 !}{8 ! ( 10 - 8 ) !} \cdot 2^{2} (- i)^{8} : 180 i^{8}

Simplify

\frac{10 !}{9 ! ( 10 - 9 ) !} \cdot 2^{1} (- i)^{9} : - 20 i^{9}

Simplify

\frac{10 !}{10 ! ( 10 - 10 ) !} \cdot 2^{0} (- i)^{10} : i^{10}

= 1024 - 5120 i + 11520 i^{2} - 15360 i^{3} + 13440 i^{4} - 8064 i^{5} + 3360 i^{6} - 960 i^{7} + 180 i^{8} - 20 i^{9} + i^{10}

Simplify

1024 - 5120 i + 11520 i^{2} - 15360 i^{3} + 13440 i^{4} - 8064 i^{5} + 3360 i^{6} - 960 i^{7} + 180 i^{8} - 20 i^{9} + i^{10} : 3116 i - 237

= 3116 i - 237

Rewrite in standard complex form:

- 237 + 3116 i

= - 237 + 3116 i

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Frequently Asked Questions (FAQ)

What is (2-i)^{10} ?
The solution to (2-i)^{10} is -237+3116i