Question

If ${\displaystyle {(x+iy)}^{\frac{1}{5}}}$ = a + ib, and u = ${\displaystyle \frac{x}{a}-\frac{y}{b}}$, then ______.

Options

• a – b is a factor of u

• a + b is a factor of x

• a + ib is a factor of y

• a – ib is a factor of a

Answer 1

If ${\displaystyle {(x+iy)}^{\frac{1}{5}}}$ = a + ib, and u = ${\displaystyle \frac{x}{a}-\frac{y}{b}}$, then a – b is a factor of u.

Explanation:

${\displaystyle {(x+iy)}^{\frac{1}{5}}}$ = (a + ib)

⇒ (x + iy) = (a + ib)⁵

x = ⁵C₀a⁵ – ⁵C₂a³b² + ⁵C₄ab⁴

y = ⁵C₁a⁴b – ⁵C₃a²b³ + ⁵C₅b⁵

u = ${\displaystyle \frac{x}{a}-\frac{y}{b}}$

= (⁵C₀a⁴ – ⁵C₂a²b² + ⁵C₄b⁴) – (⁵C₁a⁴ – ⁵C₃a²b² + ⁵C₅b⁴)

u = 4(b⁴ – a⁴)

= –4(a² + b²)(a – b)(a + b)