Question

How do you rationalize the numerator and simplify `[(1/sqrtx)+9sqrtx]/(9x+1)`?

Answer 1

The result is `sqrtx/x`.

The reason is the following:
1st) You have to rationalize `1/sqrtx`. This is done by multiplying both numerator and denominator by `sqrtx`. By doing this, you obtain the following: `((1/sqrtx) + 9sqrtx)/(9x+1) = ((sqrtx/x)+9sqrtx)/(9x+1)`.
2nd) Now, you make "x" the common denominator of the numerator as follows:
`((sqrtx/x)+9sqrtx)/(9x+1) = ((sqrtx+9xsqrtx)/x)/(9x+1)`.
3rd) Now, you pass the intermediate "x" to the denominator:
`((sqrtx+9xsqrtx)/x)/(9x+1) = (sqrtx+9xsqrtx)/(x(9x+1))`.
4th) Now, you take common factor `sqrtx` from the numerator:
`(sqrtx+9xsqrtx)/(x(9x+1)) = (sqrtx(9x+1))/(x(9x+1)`.
5th) And, finally, you simplify the factor (9x+1) appearing both in the numerator and the denominator:
`(sqrtx(cancel(9x+1)))/(x(cancel(9x+1))) = sqrtx/x`.