How do you rationalize the denominator and simplify #(x - 42) / (sqrtx+7) - 7#?

Question

Hazel Anglin · Accepted Answer

See a solution process below:

To rationalize the denominator multiply the expression by #(7 - sqrt(x))/(7 - sqrt(x))# #(7 - sqrt(x))/(7 - sqrt(x)) xx (x - 42)/(sqrt(x) + 7) - 7 =># #(7x - 294 - xsqrt(x) + 42sqrt(x))/(7sqrt(x) + 49 - x - 7sqrt(x)) - 7 =># #(7x - 294 - xsqrt(x) + 42sqrt(x))/(49 - x) - 7# To subtract the #7# we need to put it over a common denominator: #(7x - 294 - xsqrt(x) + 42sqrt(x))/(49 - x) - (7 xx (49 - x)/(49 - x)) =># #(7x - 294 - xsqrt(x) + 42sqrt(x))/(49 - x) - (343 - 7x)/(49 - x) =># #(7x - 294 - xsqrt(x) + 42sqrt(x) - 343 + 7x)/(49 - x) =># #(7x + 7x - xsqrt(x) + 42sqrt(x) - 294 - 343)/(49 - x) =># #(14x + (42 - x)sqrt(x) - 637)/(49 - x)#

Genesis Allen · Answer

undefined To rationalize the denominator and simplify the expression (x - 42) / (sqrt(x+7) - 7), you can multiply both the numerator and denominator by the conjugate of the denominator, which is (sqrt(x+7) + 7). This will eliminate the square root in the denominator. The simplified expression will be (x - 42) * (sqrt(x+7) + 7) / ((sqrt(x+7))^2 - 7^2).