What is (sqrt(2)-i)^4 ?

The solution to (sqrt(2)-i)^4 is -7-4sqrt(2)i

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

simplify (sqrt(2)-i)^4

simplify

(2 - i)^{4}

- 7 - 42 i

Solution steps

(2 - i)^{4}

Apply binomial theorem:

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

a = 2, b = - i

= i = 0 \sum 4 (i 4) (2)^{(4 - i)} (- i)^{i}

Expand summation

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

Simplify

\frac{4 !}{0 ! ( 4 - 0 ) !} (2)^{4} (- i)^{0} : 4

Simplify

\frac{4 !}{1 ! ( 4 - 1 ) !} (2)^{3} (- i)^{1} : - 82 i

Simplify

\frac{4 !}{2 ! ( 4 - 2 ) !} (2)^{2} (- i)^{2} : 12 i^{2}

Simplify

\frac{4 !}{3 ! ( 4 - 3 ) !} (2)^{1} (- i)^{3} : - 42 i^{3}

Simplify

\frac{4 !}{4 ! ( 4 - 4 ) !} (2)^{0} (- i)^{4} : i^{4}

= 4 - 82 i + 12 i^{2} - 42 i^{3} + i^{4}

Simplify

4 - 82 i + 12 i^{2} - 42 i^{3} + i^{4} : - 42 i - 7

= - 42 i - 7

Rewrite in standard complex form:

- 7 - 42 i

= - 7 - 42 i

\frac{( x + 6 ) ^{2}}{2}

\frac{( - 3 x - 8 )}{( s + 3 ) ^{2}}

\frac{6 x}{( x ^{2} - 1 ) ^{2}}

\frac{x ^{3} - ( 2 a + 1 ) x ^{2} + ( a ^{2} - 2 ) x}{x - a}

x ln (x - 5)