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Popular >

simplify (6+14i)^4

Solution

simplify

(6 + 14 i)^{4}

Solution

- 2624 - 53760 i

Solution steps

(6 + 14 i)^{4}

Apply binomial theorem:

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

a = 6, b = 14 i

= i = 0 \sum 4 (i 4) \cdot 6^{(4 - i)} (14 i)^{i}

Expand summation

= \frac{4 !}{0 ! ( 4 - 0 ) !} \cdot 6^{4} (14 i)^{0} + \frac{4 !}{1 ! ( 4 - 1 ) !} \cdot 6^{3} (14 i)^{1} + \frac{4 !}{2 ! ( 4 - 2 ) !} \cdot 6^{2} (14 i)^{2} + \frac{4 !}{3 ! ( 4 - 3 ) !} \cdot 6^{1} (14 i)^{3} + \frac{4 !}{4 ! ( 4 - 4 ) !} \cdot 6^{0} (14 i)^{4}

Simplify

\frac{4 !}{0 ! ( 4 - 0 ) !} \cdot 6^{4} (14 i)^{0} : 1296

Simplify

\frac{4 !}{1 ! ( 4 - 1 ) !} \cdot 6^{3} (14 i)^{1} : 12096 i

Simplify

\frac{4 !}{2 ! ( 4 - 2 ) !} \cdot 6^{2} (14 i)^{2} : 42336 i^{2}

Simplify

\frac{4 !}{3 ! ( 4 - 3 ) !} \cdot 6^{1} (14 i)^{3} : 65856 i^{3}

Simplify

\frac{4 !}{4 ! ( 4 - 4 ) !} \cdot 6^{0} (14 i)^{4} : 38416 i^{4}

= 1296 + 12096 i + 42336 i^{2} + 65856 i^{3} + 38416 i^{4}

Simplify

1296 + 12096 i + 42336 i^{2} + 65856 i^{3} + 38416 i^{4} : - 53760 i - 2624

= - 53760 i - 2624

Rewrite in standard complex form:

- 2624 - 53760 i

= - 2624 - 53760 i

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Frequently Asked Questions (FAQ)

What is (6+14i)^4 ?
The solution to (6+14i)^4 is -2624-53760i