A simple triangle

Looking at the given ratios of sine, we can see of applying sine rule,

$\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}$

And because $\sin A : \sin B = 5:7$ , we can conclude that $a:b = 5:7$ and as we have $c=18-a-b$ .

From the given two values of sine, $sin^{-1} \frac{2\sqrt{6}}{7} + \sin^{-1} \frac{2\sqrt{6}}{5} = 44.42^\circ +78.46^\circ = 180^\circ - 57.12^\circ$

And this you have all three angles, which are the angles of the triangle having sides in the ratio $5:6:7$ , thus the sides of the triangle are directly $5,6$ and $7$ units.

Knowing the sides, now apply the Heron's formula which is $A=\sqrt{s(s-a)(s-b)(s-c)}$ ... .... where $a,b,c$ are sides of triangle and $s$ is semiperimeter $(\frac{5+6+7}{2} = 9)$ , thus area is $A=\sqrt{9(9-5)(9-6)(9-6)} = \sqrt{9\times 4\times 3\times 2} = 6\sqrt{6}$

Hence area is $(6)^{\dfrac{3}{2}} \implies (A)^\dfrac{2}{3} = \boxed{6}$

First find $\sin(C)$ using the double angle sum and the Pythagorean identity: $\sin(C)=\sin(180°-A-B)=\sin(A+B)=\sin(A)\cos(B)+\cos(A)\sin(B)$ $=\dfrac{2\sqrt{6}}{7} \times \dfrac{1}{5}+\dfrac{5}{7} \times \dfrac{2\sqrt{6}}{5}=\dfrac{12\sqrt{6}}{35}$

Now we have $a+b+c=18$ and the equations of sine rule: $\dfrac{a}{\sin(A)}=\dfrac{b}{\sin(B)}=\dfrac{c}{\sin(C)}$ If we solve for $a$ , $b$ and $c$ , we get: $a=\dfrac{18 \sin(A)}{\sin(A)+\sin(B)+\sin(C)}=5$ $b=\dfrac{18 \sin(B)}{\sin(A)+\sin(B)+\sin(C)}=7$ $c=\dfrac{18 \sin(C)}{\sin(A)+\sin(B)+\sin(C)}=6$

Finally, we only need two sides and the angle between them to find its area by sine rule, let's choose $a$ , $b$ and the angle $C$ : $x=\dfrac{ab\sin(C)}{2}=\dfrac{5 \times 7 \times 12\sqrt{6}}{70}=6\sqrt{6}$

And the answer we want: $x^{\dfrac{2}{3}}=(6\sqrt{6})^{\dfrac{2}{3}}=\boxed{6}$

It is very easy. Imagine there is a perpendicular line BC that form a right triangle ABC. Sin(A)=opposite/hypotenuse=BC/AC. So, AC =7. Then, imagine there is another perpendicular line AC.Sin(B)=AC/AB. AB=5. The perimeter is 18. Then minus. When you get all lines, use Heron's formula. Finally, square it and cube root.