If `S=sum_(n=1)^(9999)(1)/((sqrtn+sqrt(n+1))(root4(n)+root4(n+1)))`, then the value of S is equal to

Options

A. 9
B. 99
C. 999
D. 9999

Answer

To solve the problem, we need to evaluate the sum

\[ S = \sum_{n=1}^{9999} \frac{1}{(\sqrt{n} + \sqrt{n+1})(\sqrt[4]{n} + \sqrt[4]{n+1})} \]

### Step 1: Rationalize the Denominator

We can simplify the expression in the denominator. Notice that we can multiply and divide by the conjugate of the fourth root terms:

\[ \sqrt[4]{n} - \sqrt[4]{n+1} \]

Thus, we rewrite \( S \):

\[ S = \sum_{n=1}^{9999} \frac{\sqrt[4]{n} - \sqrt[4]{n+1}}{(\sqrt{n} + \sqrt{n+1})(\sqrt[4]{n} + \sqrt[4]{n+1})(\sqrt[4]{n} - \sqrt[4]{n+1})} \]

### Step 2: Simplify the Denominator

Using the identity \( (a+b)(a-b) = a^2 - b^2 \):

\[ (\sqrt[4]{n} + \sqrt[4]{n+1})(\sqrt[4]{n} - \sqrt[4]{n+1}) = (\sqrt[4]{n})^2 - (\sqrt[4]{n+1})^2 = \sqrt{n} - \sqrt{n+1} \]

Now, substituting this back into our expression for \( S \):

\[ S = \sum_{n=1}^{9999} \frac{\sqrt[4]{n} - \sqrt[4]{n+1}}{(\sqrt{n} + \sqrt{n+1})(\sqrt{n} - \sqrt{n+1})} \]

### Step 3: Further Simplification

The denominator simplifies to:

\[ (\sqrt{n} + \sqrt{n+1})(\sqrt{n} - \sqrt{n+1}) = n - (n + 1) = -1 \]

Thus, we have:

\[ S = \sum_{n=1}^{9999} -(\sqrt[4]{n} - \sqrt[4]{n+1}) = -\left(\sqrt[4]{1} - \sqrt[4]{10000}\right) \]

### Step 4: Evaluate the Sum

Now we can evaluate the sum:

\[ S = -(\sqrt[4]{1} - \sqrt[4]{10000}) = -\left(1 - 10\right) = -(-9) = 9 \]

### Conclusion

Thus, the value of \( S \) is:

\[ \boxed{9} \]