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Popular >

simplify (3+i)^8

Solution

simplify

(3 + i)^{8}

Solution

- 8432 + 5376 i

Solution steps

(3 + i)^{8}

Apply binomial theorem:

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

a = 3, b = i

= i = 0 \sum 8 (i 8) \cdot 3^{(8 - i)} i^{i}

Expand summation

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

Simplify

\frac{8 !}{0 ! ( 8 - 0 ) !} \cdot 3^{8} i^{0} : 6561

Simplify

\frac{8 !}{1 ! ( 8 - 1 ) !} \cdot 3^{7} i^{1} : 17496 i

Simplify

\frac{8 !}{2 ! ( 8 - 2 ) !} \cdot 3^{6} i^{2} : 20412 i^{2}

Simplify

\frac{8 !}{3 ! ( 8 - 3 ) !} \cdot 3^{5} i^{3} : 13608 i^{3}

Simplify

\frac{8 !}{4 ! ( 8 - 4 ) !} \cdot 3^{4} i^{4} : 5670 i^{4}

Simplify

\frac{8 !}{5 ! ( 8 - 5 ) !} \cdot 3^{3} i^{5} : 1512 i^{5}

Simplify

\frac{8 !}{6 ! ( 8 - 6 ) !} \cdot 3^{2} i^{6} : 252 i^{6}

Simplify

\frac{8 !}{7 ! ( 8 - 7 ) !} \cdot 3^{1} i^{7} : 24 i^{7}

Simplify

\frac{8 !}{8 ! ( 8 - 8 ) !} \cdot 3^{0} i^{8} : i^{8}

= 6561 + 17496 i + 20412 i^{2} + 13608 i^{3} + 5670 i^{4} + 1512 i^{5} + 252 i^{6} + 24 i^{7} + i^{8}

Simplify

6561 + 17496 i + 20412 i^{2} + 13608 i^{3} + 5670 i^{4} + 1512 i^{5} + 252 i^{6} + 24 i^{7} + i^{8} : 5376 i - 8432

= 5376 i - 8432

Rewrite in standard complex form:

- 8432 + 5376 i

= - 8432 + 5376 i

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Frequently Asked Questions (FAQ)

What is (3+i)^8 ?
The solution to (3+i)^8 is -8432+5376i