Let's simplify the given expression step-by-step:
[tex]\[
\frac{\sqrt{3} + \sqrt{7}}{\sqrt{27} + \sqrt{63} - \sqrt{28} - \sqrt{48}}
\][/tex]
First, let's break down the terms in the denominator:
1. [tex]\(\sqrt{27}\)[/tex]:
[tex]\[
\sqrt{27} = \sqrt{9 \times 3} = 3 \sqrt{3}
\][/tex]
2. [tex]\(\sqrt{63}\)[/tex]:
[tex]\[
\sqrt{63} = \sqrt{9 \times 7} = 3 \sqrt{7}
\][/tex]
3. [tex]\(\sqrt{28}\)[/tex]:
[tex]\[
\sqrt{28} = \sqrt{4 \times 7} = 2 \sqrt{7}
\][/tex]
4. [tex]\(\sqrt{48}\)[/tex]:
[tex]\[
\sqrt{48} = \sqrt{16 \times 3} = 4 \sqrt{3}
\][/tex]
Now, substituting these simplified forms back into the denominator gives us:
[tex]\[
\sqrt{27} + \sqrt{63} - \sqrt{28} - \sqrt{48} = 3\sqrt{3} + 3\sqrt{7} - 2\sqrt{7} - 4\sqrt{3}
\][/tex]
Next, let's combine the like terms in the denominator:
[tex]\[
3\sqrt{3} - 4\sqrt{3} + 3\sqrt{7} - 2\sqrt{7}
\][/tex]
This simplifies to:
[tex]\[
(3 - 4)\sqrt{3} + (3 - 2)\sqrt{7} = -\sqrt{3} + \sqrt{7}
\][/tex]
So our simplified expression is:
[tex]\[
\frac{\sqrt{3} + \sqrt{7}}{-\sqrt{3} + \sqrt{7}}
\][/tex]
Finally, we can evaluate the numerical value of this expression. Given the resulting values:
- The numerator [tex]\(\sqrt{3} + \sqrt{7} \approx 4.3778\)[/tex]
- The denominator [tex]\(-\sqrt{3} + \sqrt{7} \approx 0.9137\)[/tex]
Thus, the expression evaluates to:
[tex]\[
\frac{4.3778}{0.9137} \approx 4.7913
\][/tex]
So, the fraction [tex]\(\frac{\sqrt{3} + \sqrt{7}}{\sqrt{27} + \sqrt{63} - \sqrt{28} - \sqrt{48}}\)[/tex] simplifies to approximately [tex]\(4.7913\)[/tex].
