Sure, let's simplify the given expressions step by step.
### Expression 1:
[tex]\[
(\sqrt{2} + \sqrt{3})(\sqrt{5} - \sqrt{7})
\][/tex]
We can use the distributive property (also known as the FOIL method for binomials):
[tex]\[
(\sqrt{2} + \sqrt{3})(\sqrt{5} - \sqrt{7}) = \sqrt{2} \cdot \sqrt{5} + \sqrt{2} \cdot (-\sqrt{7}) + \sqrt{3} \cdot \sqrt{5} + \sqrt{3} \cdot (-\sqrt{7})
\][/tex]
Simplifying each term individually:
[tex]\[
\sqrt{2} \cdot \sqrt{5} = \sqrt{10}
\][/tex]
[tex]\[
\sqrt{2} \cdot (-\sqrt{7}) = -\sqrt{14}
\][/tex]
[tex]\[
\sqrt{3} \cdot \sqrt{5} = \sqrt{15}
\][/tex]
[tex]\[
\sqrt{3} \cdot (-\sqrt{7}) = -\sqrt{21}
\][/tex]
Putting it all together, we have:
[tex]\[
\sqrt{10} - \sqrt{14} + \sqrt{15} - \sqrt{21}
\][/tex]
### Expression 2:
[tex]\[
\sqrt{10} + \sqrt{15} - \sqrt{14} - \sqrt{21}
\][/tex]
This expression is already simplified as much as possible.
### Expression 3:
[tex]\[
2\sqrt{5} - 2\sqrt{7}
\][/tex]
We can factor out the common factor 2:
[tex]\[
2(\sqrt{5} - \sqrt{7})
\][/tex]
### Expression 4:
[tex]\[
2\sqrt{5} + 3\sqrt{5} - 2\sqrt{7} - 3\sqrt{7}
\][/tex]
Combine the like terms:
[tex]\[
(2\sqrt{5} + 3\sqrt{5}) + (-2\sqrt{7} - 3\sqrt{7})
\][/tex]
[tex]\[
5\sqrt{5} - 5\sqrt{7}
\][/tex]
Combining all the simplified expressions, we get:
1. [tex]\(\sqrt{10} - \sqrt{14} + \sqrt{15} - \sqrt{21}\)[/tex]
2. [tex]\(-\sqrt{21} - \sqrt{14} + \sqrt{10} + \sqrt{15}\)[/tex]
3. [tex]\(2(\sqrt{5} - \sqrt{7})\)[/tex]
4. [tex]\(5\sqrt{5} - 5\sqrt{7}\)[/tex]
Therefore, the final answers are:
1. [tex]\((\sqrt{2} + \sqrt{3})(\sqrt{5} - \sqrt{7}) = \sqrt{10} - \sqrt{14} + \sqrt{15} - \sqrt{21}\)[/tex]
2. [tex]\(\sqrt{10} + \sqrt{15} - \sqrt{14} - \sqrt{21}\)[/tex]
3. [tex]\(2\sqrt{5} - 2\sqrt{7}\)[/tex]
4. [tex]\(2\sqrt{5} + 3\sqrt{5} - 2\sqrt{7} - 3\sqrt{7} = 5\sqrt{5} - 5\sqrt{7}\)[/tex]
Combining all the simplified expressions, we get the results:
((\sqrt{2} + \sqrt{3})(-\sqrt{7} + \sqrt{5}), -\sqrt{21} - \sqrt{14} + \sqrt{10} + \sqrt{15}, -2\sqrt{7} + 2\sqrt{5}, -5\sqrt{7} + 5\sqrt{5})
