Question

How to solve this? Is the question correct? ty :)

Answer 1

The proposition is true.

Explanation:

Now we proceed to demostrate that expression given is true by algebraic means:

1)
`(x^(-1)+y^(-1))/(x^(-1))+(x^(-1)-y^(-1) )/(y^(-1))` Given

2)
`(y^(-1)\cdot (x^(-1)+y^(-1))+x^(-1)\cdot (x^(-1)-y^(-1)))/(x^(-1)\cdot y^(-1))`
`(a)/(b) + (c)/(d) = (a\cdot d+b\cdot c)/(b\cdot d)`

3)
`(x^(-1)\cdot y^(-1)+y^(-2)+x^(-2)-x^(-1)\cdot y^(-1))/(x^(-1)\cdot y^(-1))` Distributive property/
`a^(b)\cdot a^(c) = a^(b+c)`

4)
`(x^(-2)+y^(-2))/((x\cdot y)^(-1))` Commutative, associative and modulative properties/Existence of additive inverse/
`a^(b)\cdot c^(b) = (a\cdot c)^(b)`

5)
`[(x\cdot y)^(-1)]^(-1)\cdot (x^(-2)+y^(-2))` Commutative property/Definition of division

6)
`(x\cdot y)\cdot (x^(-2)+y^(-2))`
`(x^(-1))^(-1)`

7)
`x^(-1)\cdot y + x\cdot y^(-1)` Distributive property/Associative property/
`a^(b)\cdot a^(c) = a^(b+c)`

8)
`(x^(-1)\cdot y^(-1))\cdot y^(2) + (x^(-1)\cdot y^(-1))\cdot x^(2)` Modulative property/Existence of additive inverse/
`a^(b)\cdot a^(c) = a^(b+c)`

9)
`(x^(-1)\cdot y^(-1))\cdot (x^(2)+y^(2))` Distributive property

10)
`(x\cdot y)^(-1)\cdot (x^(2)+y^(2))`
`a^(b)\cdot c^(b) = (a\cdot c)^(b)`

11)
`(x^(2)+y^(2))/(x\cdot y)` Commutative property/Definition of division/Result