To solve the expression [tex]\(\sqrt{2}(\sqrt{6} - 4\sqrt{10})\)[/tex], let's go through each step in detail:
1. Distribute [tex]\(\sqrt{2}\)[/tex] across the terms inside the parentheses:
[tex]\[\sqrt{2} \cdot \sqrt{6} - \sqrt{2} \cdot 4\sqrt{10}\][/tex]
2. Simplify each term separately:
- For the first term, [tex]\(\sqrt{2} \cdot \sqrt{6}\)[/tex]:
Using the property of square roots that [tex]\(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\)[/tex], we get:
[tex]\[\sqrt{2} \cdot \sqrt{6} = \sqrt{2 \cdot 6} = \sqrt{12}\][/tex]
- For the second term, [tex]\(\sqrt{2} \cdot 4\sqrt{10}\)[/tex]:
Again, using the property of square roots,
[tex]\[\sqrt{2} \cdot 4\sqrt{10} = 4 \cdot \sqrt{2 \cdot 10} = 4 \cdot \sqrt{20}\][/tex]
3. Simplify [tex]\(\sqrt{12}\)[/tex] and [tex]\(4 \cdot \sqrt{20}\)[/tex]:
- [tex]\(\sqrt{12}\)[/tex]:
[tex]\[\sqrt{12} = 2\sqrt{3}\][/tex]
- [tex]\(4 \cdot \sqrt{20}\)[/tex]:
[tex]\[\sqrt{20} = 2\sqrt{5} \][/tex]
Therefore,
[tex]\[4 \cdot \sqrt{20} = 4 \cdot 2\sqrt{5} = 8\sqrt{5}\][/tex]
4. Construct the simplified expression:
Substituting these simplified values back, we get:
[tex]\[\sqrt{2}(\sqrt{6} - 4\sqrt{10}) = 2\sqrt{3} - 8\sqrt{5}\][/tex]
In numerical form, evaluation of these expressions yields:
- [tex]\(2\sqrt{3} \approx 3.4641016151377544\)[/tex]
- [tex]\(8\sqrt{5} \approx 17.88854381999832\)[/tex]
Therefore, putting it together:
[tex]\[ 2\sqrt{3} - 8\sqrt{5} \approx 3.4641016151377544 - 17.88854381999832 = -14.424442204860565 \][/tex]
So, the final simplified answer is:
[tex]\[\sqrt{2}(\sqrt{6} - 4\sqrt{10}) \approx -14.424442204860565\][/tex]
