To simplify the expression [tex]\(3 \sqrt{5} - 2 \sqrt{7} + \sqrt{45} - \sqrt{28}\)[/tex], follow these steps:
1. Simplify [tex]\(\sqrt{45}\)[/tex]:
[tex]\[
\sqrt{45} = \sqrt{9 \cdot 5} = \sqrt{9} \cdot \sqrt{5} = 3 \sqrt{5}
\][/tex]
2. Simplify [tex]\(\sqrt{28}\)[/tex]:
[tex]\[
\sqrt{28} = \sqrt{4 \cdot 7} = \sqrt{4} \cdot \sqrt{7} = 2 \sqrt{7}
\][/tex]
3. Substitute the simplified radicals back into the original expression:
[tex]\[
3 \sqrt{5} - 2 \sqrt{7} + 3 \sqrt{5} - 2 \sqrt{7}
\][/tex]
4. Combine like terms:
[tex]\[
(3 \sqrt{5} + 3 \sqrt{5}) + (-2 \sqrt{7} - 2 \sqrt{7})
\][/tex]
Combine the terms with [tex]\(\sqrt{5}\)[/tex]:
[tex]\[
3 \sqrt{5} + 3 \sqrt{5} = 6 \sqrt{5}
\][/tex]
Combine the terms with [tex]\(\sqrt{7}\)[/tex]:
[tex]\[
-2 \sqrt{7} - 2 \sqrt{7} = -4 \sqrt{7}
\][/tex]
5. Write the combined simplified terms together:
[tex]\[
6 \sqrt{5} - 4 \sqrt{7}
\][/tex]
Therefore, the simplified form of the expression [tex]\(3 \sqrt{5} - 2 \sqrt{7} + \sqrt{45} - \sqrt{28}\)[/tex] is:
[tex]\[
6 \sqrt{5} - 4 \sqrt{7}
\][/tex]
