What is expand (9+i)^5 ?

The solution to expand (9+i)^5 is 51804+31996i

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expand (9+i)^5

expand

(9 + i)^{5}

51804 + 31996 i

Solution steps

(9 + i)^{5}

Apply binomial theorem:

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

a = 9, b = i

= i = 0 \sum 5 (i 5) \cdot 9^{(5 - i)} i^{i}

Expand summation

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

Simplify

\frac{5 !}{0 ! ( 5 - 0 ) !} \cdot 9^{5} i^{0} : 59049

Simplify

\frac{5 !}{1 ! ( 5 - 1 ) !} \cdot 9^{4} i^{1} : 32805 i

Simplify

\frac{5 !}{2 ! ( 5 - 2 ) !} \cdot 9^{3} i^{2} : 7290 i^{2}

Simplify

\frac{5 !}{3 ! ( 5 - 3 ) !} \cdot 9^{2} i^{3} : 810 i^{3}

Simplify

\frac{5 !}{4 ! ( 5 - 4 ) !} \cdot 9^{1} i^{4} : 45 i^{4}

Simplify

\frac{5 !}{5 ! ( 5 - 5 ) !} \cdot 9^{0} i^{5} : i^{5}

= 59049 + 32805 i + 7290 i^{2} + 810 i^{3} + 45 i^{4} + i^{5}

Simplify

59049 + 32805 i + 7290 i^{2} + 810 i^{3} + 45 i^{4} + i^{5} : 31996 i + 51804

= 31996 i + 51804

Rewrite in standard complex form:

51804 + 31996 i

= 51804 + 31996 i

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