Total distance around a shape
The dashed line is because the sign is greater than to be a solid line it has to be equal to or greater than and to the shading means any point on there works for the equation easiest way to test where you should shade is by plugging into the equation the point (0,0)
5 * 10 raise to power of - 4
For this case we have the following expression:
![2(\sqrt[4]{16x})-2(\sqrt[4]{2y})+3(\sqrt[4]{81x})-4(\sqrt[4]{32y})](https://tex.z-dn.net/?f=2%28%5Csqrt%5B4%5D%7B16x%7D%29-2%28%5Csqrt%5B4%5D%7B2y%7D%29%2B3%28%5Csqrt%5B4%5D%7B81x%7D%29-4%28%5Csqrt%5B4%5D%7B32y%7D%29)
Rewriting the numbers within the roots we have:
![2(\sqrt[4]{2*2*2*2x})-2(\sqrt[4]{2y})+3(\sqrt[4]{3*3*3*3x})-4(\sqrt[4]{2*2*2*2*2y})](https://tex.z-dn.net/?f=2%28%5Csqrt%5B4%5D%7B2%2A2%2A2%2A2x%7D%29-2%28%5Csqrt%5B4%5D%7B2y%7D%29%2B3%28%5Csqrt%5B4%5D%7B3%2A3%2A3%2A3x%7D%29-4%28%5Csqrt%5B4%5D%7B2%2A2%2A2%2A2%2A2y%7D%29)
Then, by properties of powers we have:
![2(\sqrt[4]{2^4x})-2(\sqrt[4]{2y})+3(\sqrt[4]{3^4x})-4(\sqrt[4]{2^42y})](https://tex.z-dn.net/?f=2%28%5Csqrt%5B4%5D%7B2%5E4x%7D%29-2%28%5Csqrt%5B4%5D%7B2y%7D%29%2B3%28%5Csqrt%5B4%5D%7B3%5E4x%7D%29-4%28%5Csqrt%5B4%5D%7B2%5E42y%7D%29)
Then, by radical properties we have:
![2(2\sqrt[4]{x})-2(\sqrt[4]{2y})+3(3\sqrt[4]{x})-4(2\sqrt[4]{2y})](https://tex.z-dn.net/?f=2%282%5Csqrt%5B4%5D%7Bx%7D%29-2%28%5Csqrt%5B4%5D%7B2y%7D%29%2B3%283%5Csqrt%5B4%5D%7Bx%7D%29-4%282%5Csqrt%5B4%5D%7B2y%7D%29)
Rewriting the expression we have:
![4\sqrt[4]{x}-2\sqrt[4]{2y}+9\sqrt[4]{x}-8\sqrt[4]{2y}](https://tex.z-dn.net/?f=4%5Csqrt%5B4%5D%7Bx%7D-2%5Csqrt%5B4%5D%7B2y%7D%2B9%5Csqrt%5B4%5D%7Bx%7D-8%5Csqrt%5B4%5D%7B2y%7D)
Finally, adding similar terms we have:
![(4+9)\sqrt[4]{x}-(2+8)\sqrt[4]{2y}](https://tex.z-dn.net/?f=%284%2B9%29%5Csqrt%5B4%5D%7Bx%7D-%282%2B8%29%5Csqrt%5B4%5D%7B2y%7D)
![13\sqrt[4]{x}-10\sqrt[4]{2y}](https://tex.z-dn.net/?f=13%5Csqrt%5B4%5D%7Bx%7D-10%5Csqrt%5B4%5D%7B2y%7D)
Answer:
The simplified form of the expression is:
![13\sqrt[4]{x}-10\sqrt[4]{2y}](https://tex.z-dn.net/?f=13%5Csqrt%5B4%5D%7Bx%7D-10%5Csqrt%5B4%5D%7B2y%7D)