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

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simplify (x-iy+1)^3

Solution

simplify

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

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

Solution steps

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

Apply exponent rule:

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

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

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

Apply exponent rule:

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

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

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

Expand

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

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

Distribute parentheses

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

Simplify

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

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

Rewrite in standard complex form:

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

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

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

(7 x - \frac{1}{3}) (7 x + \frac{1}{3})

\frac{e ^{(- 3 s) (4 - 8 s)}}{s ^{2} + 64}

(- 1 + i 3)^{4}

(x + i)^{4}

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