(√7+3-√21/√7+√3):(7/√21+3+3/√21-7-10/√21

To simplify the given expression \((\sqrt{7} + 3 - \frac{\sqrt{21}}{\sqrt{7}} + \sqrt{3}) : (\frac{7}{\sqrt{21}} + 3 + \frac{3}{\sqrt{21}} - 7 - \frac{10}{\sqrt{21}})\), we need to break it down step by step.

First, let's simplify the numerator:

\[
\sqrt{7} + 3 - \frac{\sqrt{21}}{\sqrt{7}} + \sqrt{3}
\]

Notice that \(\frac{\sqrt{21}}{\sqrt{7}}\) simplifies to \(\sqrt{\frac{21}{7}} = \sqrt{3}\). So the numerator becomes:

\[
\sqrt{7} + 3 - \sqrt{3} + \sqrt{3}
\]

The \(-\sqrt{3}\) and \(+\sqrt{3}\) cancel each other out, leaving us with:

\[
\sqrt{7} + 3
\]

Next, let's simplify the denominator:

\[
\frac{7}{\sqrt{21}} + 3 + \frac{3}{\sqrt{21}} - 7 - \frac{10}{\sqrt{21}}
\]

First, simplify the fractions involving \(\sqrt{21}\):

\[
\frac{7}{\sqrt{21}} = \frac{7}{\sqrt{7 \cdot 3}} = \frac{7}{\sqrt{7} \cdot \sqrt{3}} = \frac{7 \cdot \sqrt{3}}{7} = \sqrt{3}
\]

\[
\frac{3}{\sqrt{21}} = \frac{3}{\sqrt{7 \cdot 3}} = \frac{3}{\sqrt{7} \cdot \sqrt{3}} = \frac{3 \cdot \sqrt{3}}{7}
\]

\[
\frac{10}{\sqrt{21}} = \frac{10}{\sqrt{7 \cdot 3}} = \frac{10}{\sqrt{7} \cdot \sqrt{3}} = \frac{10 \cdot \sqrt{3}}{7}
\]

So the denominator becomes:

\[
\sqrt{3} + 3 + \frac{3 \cdot \sqrt{3}}{7} - 7 - \frac{10 \cdot \sqrt{3}}{7}
\]

Combine the terms involving \(\sqrt{3}\):

\[
\sqrt{3} + 3 - 7 + \left(\frac{3 \cdot \sqrt{3}}{7} - \frac{10 \cdot \sqrt{3}}{7}\right)
\]

\[
\sqrt{3} + 3 - 7 + \left(\frac{3 \cdot \sqrt{3} - 10 \cdot \sqrt{3}}{7}\right)
\]

\[
\sqrt{3} + 3 - 7 + \left(\frac{-7 \cdot \sqrt{3}}{7}\right)
\]

\[
\sqrt{3} + 3 - 7 - \sqrt{3}
\]

The \(\sqrt{3}\) and \(-\sqrt{3}\) cancel each other out, leaving us with:

\[
3 - 7 = -4
\]

Now, we have the simplified numerator and denominator:

\[
\frac{\sqrt{7} + 3}{-4}
\]

So the final simplified expression is:

\[
\frac{\sqrt{7} + 3}{-4} = -\frac{\sqrt{7} + 3}{4}
\]