Respuesta :

Answer:

[tex]\frac{1}{3x+52}[/tex]

Step-by-step explanation:

We begin with the expression

[tex]\frac{\frac{1}{x^2+51x+50} }{\frac{2}{x+50}+\frac{1}{x+1} }[/tex]

First, we can split this into the product of two fractions

[tex]\frac{1}{x^2+51x+50} *(\frac{1}{\frac{2}{x+50}+\frac{1}{x+1}} )[/tex]

Now, we can factor the denominator of the first fraction.

[tex]\frac{1}{(x+1)(x+50)} *(\frac{1}{\frac{2}{x+50}+\frac{1}{x+1}} )[/tex]

Next, we need to create a common denominator for the second fraction

[tex]\frac{2}{x+50}+\frac{1}{x+1}\\\\\frac{2}{x+50}*\frac{x+1}{x+1} +\frac{1}{x+1}*\frac{x+50}{x+50} \\\\\frac{2x+2}{(x+1)(x+50)} + \frac{x+50}{(x+1)(x+50)}\\\\ \frac{3x+52}{(x+1)(x+50)}[/tex]

Now to return this portion to the denominator of the second fraction.

[tex]\frac{1}{(x+1)(x+50)} *(\frac{1}{\frac{3x+52}{(x+1)(x+50)} } )[/tex]

Dividing by a fraction is the same as multiplying as the reciprocal of the fraction, so the second fraction becomes

[tex]\frac{1}{(x+1)(x+50)} *\frac{(x+1)(x+50)}{3x+52}[/tex]

Now to reduce this expression by canceling out [tex](x+1)(x+5)[/tex]

[tex]\frac{(x+1)(x+50)}{(x+1)(x+50)(3x+52)}\\\\\frac{1}{3x+52}[/tex]