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Popular >

simplify (3+i)^4

Solution

simplify

(3 + i)^{4}

Solution

28 + 96 i

Solution steps

(3 + i)^{4}

Apply binomial theorem:

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

a = 3, b = i

= i = 0 \sum 4 (i 4) \cdot 3^{(4 - i)} i^{i}

Expand summation

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

Simplify

\frac{4 !}{0 ! ( 4 - 0 ) !} \cdot 3^{4} i^{0} : 81

Simplify

\frac{4 !}{1 ! ( 4 - 1 ) !} \cdot 3^{3} i^{1} : 108 i

Simplify

\frac{4 !}{2 ! ( 4 - 2 ) !} \cdot 3^{2} i^{2} : 54 i^{2}

Simplify

\frac{4 !}{3 ! ( 4 - 3 ) !} \cdot 3^{1} i^{3} : 12 i^{3}

Simplify

\frac{4 !}{4 ! ( 4 - 4 ) !} \cdot 3^{0} i^{4} : i^{4}

= 81 + 108 i + 54 i^{2} + 12 i^{3} + i^{4}

Simplify

81 + 108 i + 54 i^{2} + 12 i^{3} + i^{4} : 96 i + 28

= 96 i + 28

Rewrite in standard complex form:

28 + 96 i

= 28 + 96 i

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Frequently Asked Questions (FAQ)

What is (3+i)^4 ?
The solution to (3+i)^4 is 28+96i