Simplify

Problem

(√(,x+h+5)−√(,x+5))/h

Solution

1. Rationalize the numerator by multiplying both the numerator and the denominator by the conjugate of the numerator, which is √(,x+h+5)+√(,x+5)

(√(,x+h+5)−√(,x+5))/h⋅(√(,x+h+5)+√(,x+5))/(√(,x+h+5)+√(,x+5))

2. Expand the numerator using the difference of squares formula, (a−b)*(a+b)=a2−b2

((√(,x+h+5))2−(√(,x+5))2)/(h*(√(,x+h+5)+√(,x+5)))

3. Simplify the terms in the numerator by removing the square roots.

((x+h+5)−(x+5))/(h*(√(,x+h+5)+√(,x+5)))

4. Combine like terms in the numerator to isolate h

(x+h+5−x−5)/(h*(√(,x+h+5)+√(,x+5)))

h/(h*(√(,x+h+5)+√(,x+5)))

5. Divide the numerator and denominator by h to reach the simplest form.

1/(√(,x+h+5)+√(,x+5))

Final Answer

(√(,x+h+5)−√(,x+5))/h=1/(√(,x+h+5)+√(,x+5))

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