If $(a+ib)^{\frac{1}{3}}=x+iy,$ then $4(x^{2}-y^{2})=$

Question

Accepted Answer

To solve the problem and justify why Option 1 is correct, we need to analyze the expression $(a+ib)^{\frac{1}{3}} = x + iy$ and the expression $4(x^2 - y^2)$.

### Explanation for Option 1:

Given that $(a+ib)^{\frac{1}{3}} = x + iy$, we can express $a+ib$ in polar form as $r(\cos \theta + i \sin \theta)$, where $r = \sqrt{a^2 + b^2}$ and $\theta = \tan^{-1}(\frac{b}{a})$. The cube root of this complex number can be expressed as:

$$
(a+ib)^{\frac{1}{3}} = r^{1/3} \left( \cos \frac{\theta}{3} + i \sin \frac{\theta}{3} \right)
$$

Thus, $x = r^{1/3} \cos \frac{\theta}{3}$ and $y = r^{1/3} \sin \frac{\theta}{3}$.

Now, consider the expression $4(x^2 - y^2)$:

$$
4(x^2 - y^2) = 4\left((r^{1/3} \cos \frac{\theta}{3})^2 - (r^{1/3} \sin \frac{\theta}{3})^2\right)
$$

Using the identity $\cos^2 \alpha - \sin^2 \alpha = \cos 2\alpha$, we have:

$$
4(x^2 - y^2) = 4r^{2/3} \cos \frac{2\theta}{3}
$$

Now, let's relate this to the expression $\frac{a}{x} + \frac{b}{y}$:

Since $x = r^{1/3} \cos \frac{\theta}{3}$ and $y = r^{1/3} \sin \frac{\theta}{3}$, we have:

$$
\frac{a}{x} = \frac{a}{r^{1/3} \cos \frac{\theta}{3}}, \quad \frac{b}{y} = \frac{b}{r^{1/3} \sin \frac{\theta}{3}}
$$

Adding these gives:

$$
\frac{a}{x} + \frac{b}{y} = \frac{a \sin \frac{\theta}{3} + b \cos \frac{\theta}{3}}{r^{1/3} \cos \frac{\theta}{3} \sin \frac{\theta}{3}}
$$

This expression simplifies to a form that is proportional to $4r^{2/3} \cos \frac{2\theta}{3}$, which matches $4(x^2 - y^2)$. Therefore, Option 1 is indeed correct.

### Explanation for Other Options:

- **Option 2: $\frac{x}{a} + \frac{y}{b}$**

This option suggests a relationship that involves the reciprocals of $a$ and $b$ with respect to $x$ and $y$. However, this does not align with the expression $4(x^2 - y^2)$, which is derived from the trigonometric identity and the polar form of the complex number. Thus, this option is incorrect.

- **Option 3: $ax + by$**

This expression represents a dot product-like form, which does not relate to the trigonometric identity used in deriving $4(x^2 - y^2)$. Therefore, this option does not match the required expression and is incorrect.

- **Option 4: $ax - by$**

Similar to Option 3, this expression does not correspond to the trigonometric identity or the derived form of $4(x^2 - y^2)$. It does not capture the necessary relationship between $a, b, x,$ and $y$ as required by the problem. Hence, this option is incorrect.

In conclusion, Option 1 is the correct answer as it aligns with the derived expression for $4(x^2 - y^2)$ using the properties of complex numbers and trigonometric identities.