To solve the expression [tex]\(\sqrt{6}(3 \sqrt{3} + \sqrt{10})\)[/tex], we can break it down into simpler parts and then simplify each part step by step.
1. Distribute [tex]\(\sqrt{6}\)[/tex] across the terms inside the parentheses:
[tex]\[\sqrt{6} \cdot (3 \sqrt{3}) + \sqrt{6} \cdot (\sqrt{10})\][/tex]
2. Simplify each term separately:
- For the first term: [tex]\(\sqrt{6} \cdot (3 \sqrt{3})\)[/tex]
We can start by multiplying the constants together and then the square roots together:
[tex]\[\sqrt{6} \cdot 3 \cdot \sqrt{3} = 3 \cdot (\sqrt{6} \cdot \sqrt{3}) = 3 \cdot \sqrt{6 \cdot 3} = 3 \cdot \sqrt{18}\][/tex]
Next, simplify [tex]\(\sqrt{18}\)[/tex]:
[tex]\(\sqrt{18} = \sqrt{9 \cdot 2} = \sqrt{9} \cdot \sqrt{2} = 3 \cdot \sqrt{2}\)[/tex]
So, the term becomes:
[tex]\(3 \cdot 3 \cdot \sqrt{2} = 9 \cdot \sqrt{2}\)[/tex]
- For the second term: [tex]\(\sqrt{6} \cdot \sqrt{10}\)[/tex]
Combine the square roots under a single square root:
[tex]\[\sqrt{6} \cdot \sqrt{10} = \sqrt{6 \cdot 10} = \sqrt{60}\][/tex]
Next, simplify [tex]\(\sqrt{60}\)[/tex]:
[tex]\(\sqrt{60} = \sqrt{4 \cdot 15} = \sqrt{4} \cdot \sqrt{15} = 2 \cdot \sqrt{15}\)[/tex]
3. Combine the simplified terms:
The expression [tex]\(\sqrt{6}(3 \sqrt{3} + \sqrt{10})\)[/tex] now becomes:
[tex]\[9 \sqrt{2} + 2 \sqrt{15}\][/tex]
4. Evaluate the approximate numerical values if needed:
- [tex]\(9 \sqrt{2} \approx 12.727922061357857\)[/tex]
- [tex]\(2 \sqrt{15} \approx 7.745966692414834\)[/tex]
Therefore, the simplified form of the expression [tex]\(\sqrt{6}(3 \sqrt{3} + \sqrt{10})\)[/tex] results in:
[tex]\[9 \sqrt{2} + 2 \sqrt{15} \approx 12.727922061357857 + 7.745966692414834\][/tex]
