Determine the value of
`1/(1+sqrt(2)) + 1/(sqrt2 + sqrt3) + 1/(sqrt3 + 2)`

Options

A. `-1`
B. 0
C. 1
D. 2

Answer

To solve the expression \( \frac{1}{1+\sqrt{2}} + \frac{1}{\sqrt{2}+\sqrt{3}} + \frac{1}{\sqrt{3}+2} \), we will rationalize each term and then combine them.

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

To rationalize this term, we multiply the numerator and denominator by the conjugate of the denominator, which is \( 1 - \sqrt{2} \):

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

Calculating the denominator:

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

Thus, we have:

\[ \frac{1-\sqrt{2}}{-1} = \sqrt{2} - 1 \]

### Step 2: Rationalize the second term \( \frac{1}{\sqrt{2}+\sqrt{3}} \)

Again, we multiply by the conjugate \( \sqrt{2} - \sqrt{3} \):

\[ \frac{1}{\sqrt{2}+\sqrt{3}} \cdot \frac{\sqrt{2}-\sqrt{3}}{\sqrt{2}-\sqrt{3}} = \frac{\sqrt{2}-\sqrt{3}}{(\sqrt{2}+\sqrt{3})(\sqrt{2}-\sqrt{3})} \]

Calculating the denominator:

\[ (\sqrt{2}+\sqrt{3})(\sqrt{2}-\sqrt{3}) = (\sqrt{2})^2 - (\sqrt{3})^2 = 2 - 3 = -1 \]

Thus, we have:

\[ \frac{\sqrt{2}-\sqrt{3}}{-1} = \sqrt{3} - \sqrt{2} \]

### Step 3: Rationalize the third term \( \frac{1}{\sqrt{3}+2} \)

We multiply by the conjugate \( \sqrt{3}-2 \):

\[ \frac{1}{\sqrt{3}+2} \cdot \frac{\sqrt{3}-2}{\sqrt{3}-2} = \frac{\sqrt{3}-2}{(\sqrt{3}+2)(\sqrt{3}-2)} \]

Calculating the denominator:

\[ (\sqrt{3}+2)(\sqrt{3}-2) = (\sqrt{3})^2 - 2^2 = 3 - 4 = -1 \]

Thus, we have:

\[ \frac{\sqrt{3}-2}{-1} = 2 - \sqrt{3} \]

### Step 4: Combine all the rationalized terms

Now we combine all three terms:

\[ (\sqrt{2} - 1) + (\sqrt{3} - \sqrt{2}) + (2 - \sqrt{3}) \]

Combining like terms:

- The \( \sqrt{2} \) terms: \( \sqrt{2} - \sqrt{2} = 0 \)
- The \( \sqrt{3} \) terms: \( \sqrt{3} - \sqrt{3} = 0 \)
- The constant terms: \( -1 + 2 = 1 \)

Thus, we have:

\[ 0 + 0 + 1 = 1 \]

### Final Answer

The value of the expression is \( 1 \).

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