Certainly! Let's solve the problem step-by-step:
Given the equation:
[tex]\[
\frac{2+\sqrt{3}}{2-\sqrt{3}} = 7 - a \sqrt{3}
\][/tex]
We need to find the value of \( a \) that satisfies this equation.
First, we rationalize the left-hand side by multiplying both the numerator and the denominator by the conjugate of the denominator \( 2 + \sqrt{3} \):
[tex]\[
\frac{2+\sqrt{3}}{2-\sqrt{3}} \cdot \frac{2+\sqrt{3}}{2+\sqrt{3}} = \frac{(2+\sqrt{3})(2+\sqrt{3})}{(2-\sqrt{3})(2+\sqrt{3})}
\][/tex]
Next, we simplify the numerator and the denominator:
For the numerator:
[tex]\[
(2+\sqrt{3})(2+\sqrt{3}) = 2^2 + 2\cdot 2\cdot\sqrt{3} + (\sqrt{3})^2 = 4 + 4\sqrt{3} + 3 = 7 + 4\sqrt{3}
\][/tex]
For the denominator:
[tex]\[
(2-\sqrt{3})(2+\sqrt{3}) = 2^2 - (\sqrt{3})^2 = 4 - 3 = 1
\][/tex]
Putting it all together, we get:
[tex]\[
\frac{7 + 4\sqrt{3}}{1} = 7 + 4\sqrt{3}
\][/tex]
So, our equation now reads:
[tex]\[
7 + 4\sqrt{3} = 7 - a\sqrt{3}
\][/tex]
To find \( a \), we equate the terms involving \(\sqrt{3}\):
[tex]\[
4\sqrt{3} = -a\sqrt{3}
\][/tex]
Dividing both sides by \(\sqrt{3}\), we obtain:
[tex]\[
4 = -a
\][/tex]
Thus, solving for \( a \):
[tex]\[
a = -4
\][/tex]
Therefore, the value of [tex]\( a \)[/tex] is [tex]\( \boxed{-4} \)[/tex].
