How do you convert $r = 4\sin (2\theta )$ to rectangular form?

Hint: In this question we will use the cartesian format structure and expand the value of $\sin 2\theta $ and use the properties of cartesian form to get the required solution.

Complete step-by-step solution:
We know that the cartesian form of the equation can be expressed in the format:
 $r(\cos \theta ,\sin \theta ) = (x,y)$
And we know that: $r = \sqrt {{x^2} + {y^2}} $ .
We also know that $\cos \theta = \dfrac{x}{r}$ and $\sin \theta = \dfrac{y}{r}$
We have the given equation as:
$r = 4\sin (2\theta )$
We know that $\sin 2\theta = 2\sin \theta \cos \theta $ therefore, on expanding the expression, we get:
$\Rightarrow$$r = 4(2\sin \theta \cos \theta )$
On opening the bracket, we get:
$\Rightarrow$$r = 8\sin \theta \cos \theta $
Now we substitute $\sin \theta = \dfrac{y}{r}$ and $\cos \theta = \dfrac{x}{r}$ , on substituting, we get:
$\Rightarrow$$r = 8 \times \dfrac{x}{r} \times \dfrac{y}{r}$
On multiplying the terms, we get:
$\Rightarrow$$r = \dfrac{{8xy}}{{{r^2}}}$
now we know that $r = \sqrt {{x^2} + {y^2}} $ therefore, \[{r^2} = {x^2} + {y^2}\]
on substituting the appropriate values in the equation, we get:
$\Rightarrow$$\sqrt {{x^2} + {y^2}} = \dfrac{{8xy}}{{{x^2} + {y^2}}}$
now on rearranging the equation, we get:
$\Rightarrow$$\sqrt {{x^2} + {y^2}} \times ({x^2} + {y^2}) = 8xy$
Now we know that the root value can be written in the form of exponent as:
$\Rightarrow$${({x^2} + {y^2})^{\dfrac{1}{2}}} \times ({x^2} + {y^2}) = 8xy$
Now using the property of exponents, we can write the term as:
$\Rightarrow$${({x^2} + {y^2})^{\dfrac{3}{2}}} = 8xy$
Now to remove the exponent which is in the form of a fraction, we will square both the sides.
On squaring both the sides, we get:
$\Rightarrow$${({x^2} + {y^2})^3} = {(8xy)^2}$
Now on squaring the terms in the right-hand side, we get:
$\Rightarrow$${({x^2} + {y^2})^3} = 64{x^2}{y^2}$

${({x^2} + {y^2})^3} = 64{x^2}{y^2}$ is the solution for the given question.

Note: It is to be remembered that square root can be written in the form of exponents in the reciprocal form. For example, a term $\sqrt a $ can be written as ${a^{\dfrac{1}{2}}}$ .
The property of exponents is used in the question which is ${a^m}{a^n} = {a^{m + n}}$ .
Another word for the cartesian format is the rectangular format, rectangular or cartesian coordinates are written in the format $(x,y)$ while on the other hand, polar coordinates are written in the form $(r,\theta )$ .