Sure! Let's simplify the given expression step by step:
Expression to simplify: \((2 \sqrt{3}-5 \sqrt{2})+(\sqrt{3}+2 \sqrt{2})\)
1. Distribute and combine like terms:
The expression can be rewritten as:
[tex]\[
(2 \sqrt{3} - 5 \sqrt{2}) + (\sqrt{3} + 2 \sqrt{2})
\][/tex]
2. Combine the terms involving \(\sqrt{3}\):
Group the terms with \(\sqrt{3}\):
[tex]\[
2 \sqrt{3} + \sqrt{3}
\][/tex]
These like terms can be combined by adding the coefficients:
[tex]\[
(2 + 1) \sqrt{3} = 3 \sqrt{3}
\][/tex]
3. Combine the terms involving \(\sqrt{2}\):
Group the terms with \(\sqrt{2}\):
[tex]\[
-5 \sqrt{2} + 2 \sqrt{2}
\][/tex]
These like terms can be combined by adding the coefficients:
[tex]\[
(-5 + 2) \sqrt{2} = -3 \sqrt{2}
\][/tex]
4. Combine the simplified terms:
Now combine the simplified terms from steps 2 and 3:
[tex]\[
3 \sqrt{3} - 3 \sqrt{2}
\][/tex]
Therefore, the simplified expression is:
[tex]\[
3 \sqrt{3} - 3 \sqrt{2}
\][/tex]
To verify numerically:
Upon calculating numerical values for the simplified terms:
[tex]\[
3 \sqrt{3} \approx 5.196 \quad \text{and} \quad -3 \sqrt{2} \approx -4.243
\][/tex]
Adding these values together:
[tex]\[
5.196 - 4.243 \approx 0.953
\][/tex]
So, the final result is:
[tex]\[
0.9535 \; (\text{approximately})
\][/tex]
Therefore, the simplification of [tex]\((2 \sqrt{3}-5 \sqrt{2}) + (\sqrt{3} + 2 \sqrt{2})\)[/tex] leads us to [tex]\((3 \sqrt{3} - 3 \sqrt{2}) \approx 0.9535\)[/tex].
