The solution to (x-1+iy)^3 is (x^3-3x^2+3x+3y^2-3xy^2-1)+(3x^2y-y^3+3y-6xy)i

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simplify (x-1+iy)^3

Solution

simplify

(x - 1 + i y)^{3}

(x^{3} - 3 x^{2} + 3 x + 3 y^{2} - 3 x y^{2} - 1) + (3 x^{2} y - y^{3} + 3 y - 6 x y) i

Solution steps

(x - 1 + i y)^{3}

Apply exponent rule:

a^{b + c} = a^{b} \cdot a^{c}

(x - 1 + i y)^{3} = (x - 1 + i y) (x - 1 + i y)^{2}

= (x - 1 + i y) (x - 1 + i y)^{2}

Apply exponent rule:

a^{b + c} = a^{b} \cdot a^{c}

(x - 1 + i y)^{2} = (x - 1 + i y) (x - 1 + i y)

= (x - 1 + i y) (x - 1 + i y) (x - 1 + i y)

Expand

(x - 1 + i y) (x - 1 + i y) : x^{2} - 2 x - 2 y i + 2 x y i - y^{2} + 1

= (x^{2} - 2 x - 2 y i + 2 x y i - y^{2} + 1) (x - 1 + i y)

Distribute parentheses

= x^{2} x + x^{2} (- 1) + x^{2} i y - 2 xx - 2 x (- 1) - 2 x i y - 2 y i x - 2 y i (- 1) - 2 y ii y + 2 x y i x + 2 x y i (- 1) + 2 x y ii y - y^{2} x - y^{2} (- 1) - y^{2} i y + 1 \cdot x + 1 \cdot (- 1) + 1 i y

Simplify

x^{2} x + x^{2} (- 1) + x^{2} i y - 2 xx - 2 x (- 1) - 2 x i y - 2 y i x - 2 y i (- 1) - 2 y ii y + 2 x y i x + 2 x y i (- 1) + 2 x y ii y - y^{2} x - y^{2} (- 1) - y^{2} i y + 1 \cdot x + 1 \cdot (- 1) + 1 i y : x^{3} - 3 x^{2} + 3 x^{2} y i + 3 x - i y^{3} + 3 y i - 6 x y i + 3 y^{2} - 3 x y^{2} - 1

= x^{3} - 3 x^{2} + 3 x^{2} y i + 3 x - i y^{3} + 3 y i - 6 x y i + 3 y^{2} - 3 x y^{2} - 1

Rewrite in standard complex form:

(x^{3} - 3 x^{2} + 3 x + 3 y^{2} - 3 x y^{2} - 1) + (3 x^{2} y - y^{3} + 3 y - 6 x y) i

= (x^{3} - 3 x^{2} + 3 x + 3 y^{2} - 3 x y^{2} - 1) + (3 x^{2} y - y^{3} + 3 y - 6 x y) i

(- 2 sin (t) - t cos (t))^{2}

(x - 1 + i y)^{2}

\frac{- 2 x ^{2} - 22 x - 2}{3 ( x - 2 ) ( x + 1 ) ( x - 5 )}

(x + \frac{7}{2})^{2}

e^{y + C} ∣ y ∣^{3} (4 - 4 x + y)

What is (x-1+iy)^3 ?
The solution to (x-1+iy)^3 is (x^3-3x^2+3x+3y^2-3xy^2-1)+(3x^2y-y^3+3y-6xy)i