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Popular >

simplify (1+isqrt(3))^6

Solution

simplify

(1 + i 3)^{6}

Solution

64

Solution steps

(1 + i 3)^{6}

Apply binomial theorem:

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

a = 1, b = i 3

= i = 0 \sum 6 (i 6) \cdot 1^{(6 - i)} (i 3)^{i}

Expand summation

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

Simplify

\frac{6 !}{0 ! ( 6 - 0 ) !} \cdot 1^{6} (i 3)^{0} : 1

Simplify

\frac{6 !}{1 ! ( 6 - 1 ) !} \cdot 1^{5} (i 3)^{1} : 63 i

Simplify

\frac{6 !}{2 ! ( 6 - 2 ) !} \cdot 1^{4} (i 3)^{2} : 45 i^{2}

Simplify

\frac{6 !}{3 ! ( 6 - 3 ) !} \cdot 1^{3} (i 3)^{3} : 603 i^{3}

Simplify

\frac{6 !}{4 ! ( 6 - 4 ) !} \cdot 1^{2} (i 3)^{4} : 135 i^{4}

Simplify

\frac{6 !}{5 ! ( 6 - 5 ) !} \cdot 1^{1} (i 3)^{5} : 543 i^{5}

Simplify

\frac{6 !}{6 ! ( 6 - 6 ) !} \cdot 1^{0} (i 3)^{6} : 27 i^{6}

= 1 + 63 i + 45 i^{2} + 603 i^{3} + 135 i^{4} + 543 i^{5} + 27 i^{6}

Simplify

1 + 63 i + 45 i^{2} + 603 i^{3} + 135 i^{4} + 543 i^{5} + 27 i^{6} : 64

= 64

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Frequently Asked Questions (FAQ)

What is (1+isqrt(3))^6 ?
The solution to (1+isqrt(3))^6 is 64