Let `T = (1)/(3-sqrt(8))-(1)/(sqrt(8)-sqrt(7)) +(1)/(sqrt(7)-sqrt(6))-(1)/(sqrt(6)-sqrt(5))+(1)/(sqrt(5)+2)` then-

Options

A. `T lt 1`
B. `T = 1`
C. `1 lt T lt 2`
D. `T lt 2`

Answer

To solve the expression

\[ T = \frac{1}{3 - \sqrt{8}} - \frac{1}{\sqrt{8} - \sqrt{7}} + \frac{1}{\sqrt{7} - \sqrt{6}} - \frac{1}{\sqrt{6} - \sqrt{5}} + \frac{1}{\sqrt{5} + 2} \]

we will rationalize each term in the expression step by step.

### Step 1: Rationalize the first term \(\frac{1}{3 - \sqrt{8}}\)

Multiply the numerator and denominator by the conjugate of the denominator:

\[ \frac{1}{3 - \sqrt{8}} \cdot \frac{3 + \sqrt{8}}{3 + \sqrt{8}} = \frac{3 + \sqrt{8}}{(3 - \sqrt{8})(3 + \sqrt{8})} = \frac{3 + \sqrt{8}}{9 - 8} = 3 + \sqrt{8} \]

### Step 2: Rationalize the second term \(-\frac{1}{\sqrt{8} - \sqrt{7}}\)

Multiply the numerator and denominator by the conjugate of the denominator:

\[ -\frac{1}{\sqrt{8} - \sqrt{7}} \cdot \frac{\sqrt{8} + \sqrt{7}}{\sqrt{8} + \sqrt{7}} = -\frac{\sqrt{8} + \sqrt{7}}{(\sqrt{8} - \sqrt{7})(\sqrt{8} + \sqrt{7})} = -\frac{\sqrt{8} + \sqrt{7}}{8 - 7} = -(\sqrt{8} + \sqrt{7}) \]

### Step 3: Rationalize the third term \(\frac{1}{\sqrt{7} - \sqrt{6}}\)

Multiply the numerator and denominator by the conjugate of the denominator:

\[ \frac{1}{\sqrt{7} - \sqrt{6}} \cdot \frac{\sqrt{7} + \sqrt{6}}{\sqrt{7} + \sqrt{6}} = \frac{\sqrt{7} + \sqrt{6}}{(\sqrt{7} - \sqrt{6})(\sqrt{7} + \sqrt{6})} = \frac{\sqrt{7} + \sqrt{6}}{7 - 6} = \sqrt{7} + \sqrt{6} \]

### Step 4: Rationalize the fourth term \(-\frac{1}{\sqrt{6} - \sqrt{5}}\)

Multiply the numerator and denominator by the conjugate of the denominator:

\[ -\frac{1}{\sqrt{6} - \sqrt{5}} \cdot \frac{\sqrt{6} + \sqrt{5}}{\sqrt{6} + \sqrt{5}} = -\frac{\sqrt{6} + \sqrt{5}}{(\sqrt{6} - \sqrt{5})(\sqrt{6} + \sqrt{5})} = -\frac{\sqrt{6} + \sqrt{5}}{6 - 5} = -(\sqrt{6} + \sqrt{5}) \]

### Step 5: Rationalize the fifth term \(\frac{1}{\sqrt{5} + 2}\)

Multiply the numerator and denominator by the conjugate of the denominator:

\[ \frac{1}{\sqrt{5} + 2} \cdot \frac{\sqrt{5} - 2}{\sqrt{5} - 2} = \frac{\sqrt{5} - 2}{(\sqrt{5} + 2)(\sqrt{5} - 2)} = \frac{\sqrt{5} - 2}{5 - 4} = \sqrt{5} - 2 \]

### Step 6: Combine all the terms

Now substituting back into \(T\):

\[ T = (3 + \sqrt{8}) - (\sqrt{8} + \sqrt{7}) + (\sqrt{7} + \sqrt{6}) - (\sqrt{6} + \sqrt{5}) + (\sqrt{5} - 2) \]

### Step 7: Simplify the expression

Combining like terms:

- The \(\sqrt{8}\) cancels: \(\sqrt{8} - \sqrt{8} = 0\)
- The \(\sqrt{7}\) cancels: \(-\sqrt{7} + \sqrt{7} = 0\)
- The \(\sqrt{6}\) cancels: \(-\sqrt{6} + \sqrt{6} = 0\)
- The \(\sqrt{5}\) cancels: \(-\sqrt{5} + \sqrt{5} = 0\)

Thus, we are left with:

\[ T = 3 - 2 = 1 \]

### Final Result

The value of \(T\) is

\[ \boxed{1} \]