Simplify (1/( square root of 4+h)-1/2)/h

Problem

(1/√(,4+h)−1/2)/h

Solution

1. Find a common denominator for the terms in the numerator by multiplying the first fraction by 2/2 and the second fraction by √(,4+h)/√(,4+h)

(2−√(,4+h))/(2√(,4+h))/h

2. Rewrite the expression by multiplying the numerator by the reciprocal of the denominator h

(2−√(,4+h))/(2*h√(,4+h))

3. Rationalize the numerator by multiplying both the numerator and the denominator by the conjugate 2+√(,4+h)

((2−√(,4+h))*(2+√(,4+h)))/(2*h√(,4+h)*(2+√(,4+h)))

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

(4−(4+h))/(2*h√(,4+h)*(2+√(,4+h)))

5. Simplify the numerator by distributing the negative sign and combining like terms.

(−h)/(2*h√(,4+h)*(2+√(,4+h)))

6. Cancel the common factor h from the numerator and the denominator, assuming h≠0

(−1)/(2√(,4+h)*(2+√(,4+h)))

7. Distribute the denominator to reach the final simplified form.

(−1)/(4√(,4+h)+2*(4+h))

Final Answer

(1/√(,4+h)−1/2)/h=(−1)/(4√(,4+h)+8+2*h)

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