To find the mean of the numbers [tex]\(\sqrt{2}, \sqrt{18}, \sqrt{50}\)[/tex], we will first simplify each square root that we can, and then calculate the mean in the form of [tex]\(a \sqrt{2}\)[/tex].
1. Simplify [tex]\(\sqrt{18}\)[/tex]:
[tex]\[
\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3 \sqrt{2}
\][/tex]
2. Simplify [tex]\(\sqrt{50}\)[/tex]:
[tex]\[
\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5 \sqrt{2}
\][/tex]
3. List the simplified numbers:
[tex]\[
\sqrt{2}, \quad 3\sqrt{2}, \quad 5\sqrt{2}
\][/tex]
4. Sum these numbers:
[tex]\[
\sqrt{2} + 3\sqrt{2} + 5\sqrt{2}
\][/tex]
Combining like terms, we get:
[tex]\[
(1\sqrt{2} + 3\sqrt{2} + 5\sqrt{2}) = 9\sqrt{2}
\][/tex]
5. Calculate the mean:
[tex]\[
\text{Mean} = \frac{\text{Sum of the numbers}}{\text{Number of the terms}} = \frac{9\sqrt{2}}{3} = 3\sqrt{2}
\][/tex]
So, the mean of [tex]\(\sqrt{2}, \sqrt{18}, \sqrt{50}\)[/tex] is [tex]\(3\sqrt{2}\)[/tex].
Therefore, the value of [tex]\(a\)[/tex] is [tex]\(3\)[/tex].
