Sure! Let's solve the expression [tex]\(\frac{\sqrt{7} - 3\sqrt{2}}{2\sqrt{7} - \sqrt{3}}\)[/tex] step-by-step.
### Step 1: Rationalize the denominator
To rationalize the denominator of the given expression, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of [tex]\(2\sqrt{7} - \sqrt{3}\)[/tex] is [tex]\(2\sqrt{7} + \sqrt{3}\)[/tex].
So, we multiply [tex]\(\frac{\sqrt{7} - 3\sqrt{2}}{2\sqrt{7} - \sqrt{3}}\)[/tex] by [tex]\(\frac{2\sqrt{7} + \sqrt{3}}{2\sqrt{7} + \sqrt{3}}\)[/tex]:
[tex]\[
\frac{(\sqrt{7} - 3\sqrt{2})(2\sqrt{7} + \sqrt{3})}{(2\sqrt{7} - \sqrt{3})(2\sqrt{7} + \sqrt{3})}
\][/tex]
### Step 2: Simplify the denominator
The denominator becomes a difference of squares:
[tex]\[
(2\sqrt{7})^2 - (\sqrt{3})^2 = 4 \cdot 7 - 3 = 28 - 3 = 25
\][/tex]
So the simplified denominator is 25.
### Step 3: Expand the numerator
Next, we expand the numerator [tex]\((\sqrt{7} - 3\sqrt{2})(2\sqrt{7} + \sqrt{3})\)[/tex]:
- The first term: [tex]\(\sqrt{7} \cdot 2\sqrt{7} = 2 \cdot 7 = 14\)[/tex]
- The second term: [tex]\(\sqrt{7} \cdot \sqrt{3} = \sqrt{21}\)[/tex]
- The third term: [tex]\(-3\sqrt{2} \cdot 2\sqrt{7} = -6\sqrt{14}\)[/tex]
- The fourth term: [tex]\(-3\sqrt{2} \cdot \sqrt{3} = -3\sqrt{6}\)[/tex]
So, combining these terms, we get:
[tex]\[
14 + \sqrt{21} - 6\sqrt{14} - 3\sqrt{6}
\][/tex]
Now we have:
[tex]\[
\frac{14 + \sqrt{21} - 6\sqrt{14} - 3\sqrt{6}}{25}
\][/tex]
### Step 4: Simplifying the fraction
The numerator simplifies to [tex]\(-11.215837854037344\)[/tex]. Dividing this by the simplified denominator of 25, we get the final value of the expression:
[tex]\[
\frac{-11.215837854037344}{25} = -0.4486335141614937
\][/tex]
### Conclusion
Therefore, the value of the expression [tex]\(\frac{\sqrt{7} - 3\sqrt{2}}{2\sqrt{7} - \sqrt{3}}\)[/tex] is approximately [tex]\( -0.4486335141614937 \)[/tex].
