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Popular >

simplify (2+i)^4

Solution

simplify

(2 + i)^{4}

Solution

- 7 + 24 i

Solution steps

(2 + i)^{4}

Apply binomial theorem:

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

a = 2, b = i

= i = 0 \sum 4 (i 4) \cdot 2^{(4 - i)} i^{i}

Expand summation

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

Simplify

\frac{4 !}{0 ! ( 4 - 0 ) !} \cdot 2^{4} i^{0} : 16

Simplify

\frac{4 !}{1 ! ( 4 - 1 ) !} \cdot 2^{3} i^{1} : 32 i

Simplify

\frac{4 !}{2 ! ( 4 - 2 ) !} \cdot 2^{2} i^{2} : 24 i^{2}

Simplify

\frac{4 !}{3 ! ( 4 - 3 ) !} \cdot 2^{1} i^{3} : 8 i^{3}

Simplify

\frac{4 !}{4 ! ( 4 - 4 ) !} \cdot 2^{0} i^{4} : i^{4}

= 16 + 32 i + 24 i^{2} + 8 i^{3} + i^{4}

Simplify

16 + 32 i + 24 i^{2} + 8 i^{3} + i^{4} : 24 i - 7

= 24 i - 7

Rewrite in standard complex form:

- 7 + 24 i

= - 7 + 24 i

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Frequently Asked Questions (FAQ)

What is (2+i)^4 ?
The solution to (2+i)^4 is -7+24i