What is (1+2 i/n-2/n-2)^3 ?

The solution to (1+2 i/n-2/n-2)^3 is (-n^3-6n^2+16)/(n^3)+(16+6n^2+24n)/(n^3)i

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simplify (1+2 i/n-2/n-2)^3

simplify

(1 + 2 \frac{i}{n} - \frac{2}{n} - 2)^{3}

\frac{- n ^{3} - 6 n ^{2} + 16}{n ^{3}} + \frac{16 + 6 n ^{2} + 24 n}{n ^{3}} i

Solution steps

(1 + 2 \cdot \frac{i}{n} - \frac{2}{n} - 2)^{3}

Rewrite

1 + 2 \cdot \frac{i}{n} - \frac{2}{n} - 2

in standard complex form:

(1 - \frac{2}{n} - 2) + 2 \cdot \frac{1}{n} i

= ((1 - \frac{2}{n} - 2) + 2 \cdot \frac{1}{n} i)^{3}

Simplify

(1 - \frac{2}{n} - 2) + 2 \cdot \frac{1}{n} i : \frac{- n - 2}{n} + \frac{2}{n} i

= (\frac{- n - 2}{n} + \frac{2}{n} i)^{3}

Apply Perfect Cube Formula:

(a + b)^{3} = a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}

(\frac{- n - 2}{n} + \frac{2}{n} i)^{3} = (\frac{- n - 2}{n})^{3} + 3 (\frac{- n - 2}{n})^{2} \frac{2}{n} i + 3 \cdot \frac{- n - 2}{n} (\frac{2}{n} i)^{2} + (\frac{2}{n} i)^{3}

= (\frac{- n - 2}{n})^{3} + 3 (\frac{- n - 2}{n})^{2} \frac{2}{n} i + 3 \cdot \frac{- n - 2}{n} (\frac{2}{n} i)^{2} + (\frac{2}{n} i)^{3}

Simplify

(\frac{- n - 2}{n})^{3} + 3 (\frac{- n - 2}{n})^{2} \frac{2}{n} i + 3 \cdot \frac{- n - 2}{n} (\frac{2}{n} i)^{2} + (\frac{2}{n} i)^{3} : \frac{- n ^{3} - 6 n ^{2} + 16}{n ^{3}} + \frac{16 + 6 n ^{2} + 24 n}{n ^{3}} i

= \frac{- n ^{3} - 6 n ^{2} + 16}{n ^{3}} + \frac{16 + 6 n ^{2} + 24 n}{n ^{3}} i

x^{3} + x - 3 x^{2} - 3

5 q + 5 q

\frac{( x ^{2} + y ^{2} )}{( x ^{3} + y ^{3} )}

(x + 5)^{3} + (x - 5)^{3}

(- 5.3) y (4.3)

What is (1+2 i/n-2/n-2)^3 ?
The solution to (1+2 i/n-2/n-2)^3 is (-n^3-6n^2+16)/(n^3)+(16+6n^2+24n)/(n^3)i