To solve the expression [tex]\(-3 \times \sqrt{2} + \frac{\sqrt{2}}{4} \div \sqrt{2} + \frac{4}{5} \times \sqrt{125}\)[/tex], we will break it down into individual terms and compute each term step-by-step for clarity.
1. First Term: [tex]\(-3 \times \sqrt{2}\)[/tex]
- We multiply [tex]\(-3\)[/tex] by [tex]\(\sqrt{2}\)[/tex].
[tex]\[
-3 \times \sqrt{2} \approx -4.242640687119286
\][/tex]
2. Second Term: [tex]\(\frac{\sqrt{2}}{4} \div \sqrt{2}\)[/tex]
- First, we simplify the division inside the expression. The division of [tex]\(\frac{\sqrt{2}}{4}\)[/tex] by [tex]\(\sqrt{2}\)[/tex] can be simplified as follows:
[tex]\[
\frac{\sqrt{2}}{4} \div \sqrt{2} = \frac{\sqrt{2}}{4} \times \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{4\sqrt{2}} = \frac{1}{4} = 0.25
\][/tex]
3. Third Term: [tex]\(\frac{4}{5} \times \sqrt{125}\)[/tex]
- First, we simplify [tex]\(\sqrt{125}\)[/tex]. Since [tex]\(125\)[/tex] can be written as [tex]\(5^3\)[/tex], we have:
[tex]\[
\sqrt{125} = \sqrt{5^3} = \sqrt{5 \times 5 \times 5} = 5\sqrt{5}
\][/tex]
- Then, we multiply [tex]\(\frac{4}{5}\)[/tex] by [tex]\(5\sqrt{5}\)[/tex]:
[tex]\[
\frac{4}{5} \times 5\sqrt{5} = 4 \times \sqrt{5}\approx 8.94427190999916
\][/tex]
4. Adding the Terms Together
Now that we have each term calculated, we can add them together:
[tex]\[
-4.242640687119286 + 0.25 + 8.94427190999916 \approx 4.951631222879874
\][/tex]
Thus, the final result of the expression [tex]\( -3 \times \sqrt{2} + \frac{\sqrt{2}}{4} \div \sqrt{2} + \frac{4}{5} \times \sqrt{125} \)[/tex] is approximately [tex]\( 4.951631222879874 \)[/tex].
