To multiply and simplify the expression [tex]\(\sqrt{3}(7 \sqrt{10} + \sqrt{7})\)[/tex], follow these steps:
1. Distribute [tex]\(\sqrt{3}\)[/tex] to each term inside the parentheses:
[tex]\[
\sqrt{3} \cdot 7 \sqrt{10} + \sqrt{3} \cdot \sqrt{7}
\][/tex]
2. Multiply [tex]\(\sqrt{3}\)[/tex] with [tex]\(7 \sqrt{10}\)[/tex]:
[tex]\[
7 \cdot \sqrt{3} \cdot \sqrt{10}
\][/tex]
We can use the property of square roots [tex]\(\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}\)[/tex]:
[tex]\[
7 \cdot \sqrt{3 \cdot 10} = 7 \cdot \sqrt{30}
\][/tex]
Numerically, this gives:
[tex]\[
7 \cdot \sqrt{30} \approx 38.34057902536163
\][/tex]
3. Multiply [tex]\(\sqrt{3}\)[/tex] with [tex]\(\sqrt{7}\)[/tex]:
[tex]\[
\sqrt{3} \cdot \sqrt{7}
\][/tex]
Again, using the square root property:
[tex]\[
\sqrt{3 \cdot 7} = \sqrt{21}
\][/tex]
Numerically, this gives:
[tex]\[
\sqrt{21} \approx 4.58257569495584
\][/tex]
4. Add the results from steps 2 and 3:
[tex]\[
7 \sqrt{30} + \sqrt{21}
\][/tex]
Numerically, adding these two results:
[tex]\[
38.34057902536163 + 4.58257569495584 = 42.92315472031747
\][/tex]
Hence, the simplified expression is approximately:
[tex]\[
\boxed{42.92315472031747}
\][/tex]
