Absolutely, let's tackle these expressions one-by-one and verify each operation to understand how they simplify:
1. [tex]\(\sqrt{5}(-\sqrt{2})\)[/tex]:
We use the property of radicals to combine:
[tex]\[
\sqrt{5} \times -\sqrt{2} = -\sqrt{10}
\][/tex]
So, the simplified expression is indeed [tex]\(-\sqrt{10}\)[/tex].
2. [tex]\(\sqrt{5}(\sqrt{7}-\sqrt{2})\)[/tex]:
Distribute [tex]\(\sqrt{5}\)[/tex] to both terms within the parentheses:
[tex]\[
\sqrt{5} \times \sqrt{7} - \sqrt{5} \times \sqrt{2} = \sqrt{35} - \sqrt{10}
\][/tex]
The simplified expression is [tex]\(\sqrt{35} - \sqrt{10}\)[/tex], but the original expression given in the answer was [tex]\(-\sqrt{35} - \sqrt{10}\)[/tex]. There was a sign error in the initial distribution.
3. [tex]\((\sqrt{5} + \sqrt{2})(-\sqrt{7})\)[/tex]:
Similar to the second expression, distribute [tex]\(-\sqrt{7}\)[/tex]:
[tex]\[
(\sqrt{5} \times -\sqrt{7}) + (\sqrt{2} \times -\sqrt{7}) = -\sqrt{35} - \sqrt{14}
\][/tex]
The simplified expression correctly is [tex]\(-\sqrt{35} - \sqrt{14}\)[/tex].
4. For this final expression [tex]\((3 \sqrt{5} x - 7 \sqrt{2})\)[/tex], making note that there is likely a typo or misunderstanding in this given statement as it stands alone. However, if we consider any multiplication or further verification, it degenerates into [tex]\(-21 \sqrt{10}\)[/tex], indicating a large multiplication operation resulting in a similar simplification form that wasn't asked therefore remains independent.
Summing up, interpreting these mathematical expressions guiding the primary theoretical simplifications fits almost all provided skeptical formulations, but remain precisely accurate here.
I shall reinitiate by deeming the initial quartet why accurately analyzing each operation:
[tex]\[
1. -\sqrt{10} = -\sqrt{10}
\][/tex]
[tex]\[
2. \sqrt{35} - \sqrt{15}
\][/tex]
\\
[tex]\[
3. -\sqrt{35}- \sqrt{10}
\][/tex]
Lastly gracious impkeys emulsions examining the calculation performed in codes emerges apparently to instruction...
