Answer:
[tex]\sf \dfrac{y^2+2x^2y}{ 5xy^2 - 6x^2 } [/tex]
Step-by-step Explanation:
To simplify the given expression, let's first find a common denominator for all the fractions involved:
[tex]\sf \dfrac{\dfrac{1}{x^2} + \dfrac{2}{y}}{\dfrac{5}{x} - \dfrac{6}{y^2 }} [/tex]
To get a common denominator, we multiply each fraction by the least common multiple (LCM) of the denominators:
For [tex]\sf \dfrac{1}{x^2} [/tex] and [tex]\sf \dfrac{2}{y} [/tex], the LCM of the denominators is [tex]\sf x^2y [/tex].
For [tex]\sf \dfrac{5}{x} [/tex] and [tex]\sf \dfrac{6}{y^2} [/tex], the LCM of the denominators is [tex]\sf x \cdot y^2 [/tex].
Now, we rewrite the expression with the common denominator:
[tex]\sf \dfrac{\dfrac{1 \cdot y}{x^2 \cdot y} + \dfrac{2 \cdot x^2}{y \cdot x^2}}{\dfrac{5 \cdot y^2}{x \cdot y^2} - \dfrac{6 \cdot x}{y^2 \cdot x}} [/tex]
[tex]\sf = \dfrac{\dfrac{y}{x^2y} + \dfrac{2x^2}{xy^2}}{\dfrac{5y^2}{xy^2} - \dfrac{6x}{xy^2}} [/tex]
[tex]\sf = \dfrac{\dfrac{y + 2x^2}{x^2y}}{\dfrac{5y^2 - 6x}{xy^2}} [/tex]
Now, to divide a fraction by another fraction, we multiply the first fraction by the reciprocal of the second fraction:
[tex]\sf = \dfrac{y + 2x^2}{\cancel{x}\cdot x \cancel{y}} \cdot \dfrac{\cancel{x}\cdot \cancel{y} \cdot y}{5y^2 - 6x} [/tex]
[tex]\sf = \dfrac{(y+2x^2) \cdot y }{ x \cdot (5y^2 - 6x) } [/tex]
Distribute the bracket:
[tex]\sf = \dfrac{y^2+2x^2y}{ 5xy^2 - 6x^2 } [/tex]
So, the simplified expression is:
[tex]\sf \boxed{\boxed{ \dfrac{y^2+2x^2y}{ 5xy^2 - 6x^2 } }} [/tex]
