Let `a=(1)/(2-sqrt(3))+(1)/(3-sqrt(8))+(1)/(4-sqrt(15))` Then we have .

Options

A. `alt18"but"a ne 9`
B. `agt18`
C. `a=18`
D. `a=9`

Answer

To solve the expression \( a = \frac{1}{2 - \sqrt{3}} + \frac{1}{3 - \sqrt{8}} + \frac{1}{4 - \sqrt{15}} \), we will rationalize each term in the sum.

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

To rationalize, we multiply the numerator and the denominator by the conjugate of the denominator, which is \( 2 + \sqrt{3} \):

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

Now, simplify the denominator using the difference of squares:

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

Thus, the first term simplifies to:

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

### Step 2: Rationalize the second term \( \frac{1}{3 - \sqrt{8}} \)

Again, we multiply the numerator and the denominator by the conjugate of the denominator, \( 3 + \sqrt{8} \):

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

Now, simplify the denominator:

\[ (3 - \sqrt{8})(3 + \sqrt{8}) = 3^2 - (\sqrt{8})^2 = 9 - 8 = 1 \]

Thus, the second term simplifies to:

\[ \frac{3 + \sqrt{8}}{1} = 3 + \sqrt{8} \]

### Step 3: Rationalize the third term \( \frac{1}{4 - \sqrt{15}} \)

We multiply the numerator and the denominator by the conjugate of the denominator, \( 4 + \sqrt{15} \):

\[ \frac{1}{4 - \sqrt{15}} \cdot \frac{4 + \sqrt{15}}{4 + \sqrt{15}} = \frac{4 + \sqrt{15}}{(4 - \sqrt{15})(4 + \sqrt{15})} \]

Now, simplify the denominator:

\[ (4 - \sqrt{15})(4 + \sqrt{15}) = 4^2 - (\sqrt{15})^2 = 16 - 15 = 1 \]

Thus, the third term simplifies to:

\[ \frac{4 + \sqrt{15}}{1} = 4 + \sqrt{15} \]

### Step 4: Combine all terms

Now we combine all three simplified terms:

\[ a = (2 + \sqrt{3}) + (3 + \sqrt{8}) + (4 + \sqrt{15}) \]

Combine the constant terms and the square root terms:

\[ a = (2 + 3 + 4) + (\sqrt{3} + \sqrt{8} + \sqrt{15}) = 9 + \sqrt{3} + \sqrt{8} + \sqrt{15} \]

### Final Result

Thus, the final expression for \( a \) is:

\[ a = 9 + \sqrt{3} + \sqrt{8} + \sqrt{15} \]