Question

23 number please
thank youu​

Answer 1

Explanation:

`(\left(x^2-(1)/(y^2)\right)^x\left(x-(1)/(y)\right)^(y-x))/(\left(y^2-(1)/(x^2)\right)^y\left(y+(1)/(x)\right)^(x-y))\\\\=(\left((x^2y^2-1)/(y^2)\right)^x\left((xy)/(y)-(1)/(y)\right)^(y-x))/(\left((x^2y^2)/(x^2)\right)^y\left((xy)/(x)+(1)/(x)\right)^(x-y))`

`=(\left((x^2y^2-1)/(y^2)\right)^x\left((xy-1)/(y)\right)^y\left((xy-1)/(y)\right)^(-x))/(\left((x^2y^2)/(x^2)\right)^y\left((xy+1)/(x)\right)^x\left((xy+1)/(x)\right)^(-y))\\\\=(\left((x^2y^2-1)/(y^2)\right)^x\left((xy-1)/(y)\right)^y\left((y)/(xy-1)\right)^x)/(\left((x^2y^2-1)/(x^2)\right)^y\left((xy+1)/(x)\right)^x\left((x)/(xy+1)\right)^y)`

`=\left((x^2y^2-1)/(y^2)\cdot(y)/(xy-1)\cdot(x)/(xy+1)\right)^x\left((xy-1)/(y)\cdot(x^2)/(x^2y^2-1)\cdot(xy+1)/(x)\right)^y`

`=\left(((xy-1)(xy+1)xy)/(y^2(xy-1)(xy+1))\right)^x\left((x^2(xy-1)(xy+1))/(xy(xy-1)(xy+1))\right)^y\\\\=\left((x)/(y)\right)^x\left((x)/(y)\right)^y=\left((x)/(y)\right)^(x+y)`

Used:

`a^(-n)=\left((1)/(a)\right)^n\\\\(a\cdot b)^n=a^n\cdot b^n\\\\\left((a)/(b)\right)^n=(a^n)/(b^n)\\\\a^2-b^2=(a-b)(a+b)\\\\a^n\cdot a^m=a^(n+m)`