What is (1-isqrt(7))^6 ?

The solution to (1-isqrt(7))^6 is 288-160sqrt(7)i

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simplify (1-isqrt(7))^6

simplify

(1 - i 7)^{6}

288 - 1607 i

Solution steps

(1 - i 7)^{6}

Apply binomial theorem:

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

a = 1, b = - i 7

= i = 0 \sum 6 (i 6) \cdot 1^{(6 - i)} (- i 7)^{i}

Expand summation

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

Simplify

\frac{6 !}{0 ! ( 6 - 0 ) !} \cdot 1^{6} (- i 7)^{0} : 1

Simplify

\frac{6 !}{1 ! ( 6 - 1 ) !} \cdot 1^{5} (- i 7)^{1} : - 67 i

Simplify

\frac{6 !}{2 ! ( 6 - 2 ) !} \cdot 1^{4} (- i 7)^{2} : 105 i^{2}

Simplify

\frac{6 !}{3 ! ( 6 - 3 ) !} \cdot 1^{3} (- i 7)^{3} : - 1407 i^{3}

Simplify

\frac{6 !}{4 ! ( 6 - 4 ) !} \cdot 1^{2} (- i 7)^{4} : 735 i^{4}

Simplify

\frac{6 !}{5 ! ( 6 - 5 ) !} \cdot 1^{1} (- i 7)^{5} : - 2947 i^{5}

Simplify

\frac{6 !}{6 ! ( 6 - 6 ) !} \cdot 1^{0} (- i 7)^{6} : 343 i^{6}

= 1 - 67 i + 105 i^{2} - 1407 i^{3} + 735 i^{4} - 2947 i^{5} + 343 i^{6}

Simplify

1 - 67 i + 105 i^{2} - 1407 i^{3} + 735 i^{4} - 2947 i^{5} + 343 i^{6} : - 1607 i + 288

= - 1607 i + 288

Rewrite in standard complex form:

288 - 1607 i

= 288 - 1607 i

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