Let's simplify each given expression step-by-step.
1. Simplifying [tex]\(\sqrt{50 x^2}\)[/tex]:
[tex]\[
\sqrt{50 x^2} = \sqrt{25 \cdot 2 \cdot x^2}
\][/tex]
The expression can be further simplified using the property of square roots:
[tex]\[
\sqrt{25 \cdot 2 \cdot x^2} = \sqrt{25} \cdot \sqrt{2} \cdot \sqrt{x^2}
\][/tex]
Since [tex]\(\sqrt{25} = 5\)[/tex] and [tex]\(\sqrt{x^2} = x\)[/tex] for [tex]\(x \geq 0\)[/tex]:
[tex]\[
\sqrt{25 \cdot 2 \cdot x^2} = 5x \sqrt{2}
\][/tex]
Thus, [tex]\(b = 2\)[/tex].
2. Simplifying [tex]\(\sqrt{32 x}\)[/tex]:
[tex]\[
\sqrt{32 x} = \sqrt{16 \cdot 2 \cdot x}
\][/tex]
Similar to the previous step:
[tex]\[
\sqrt{16 \cdot 2 \cdot x} = \sqrt{16} \cdot \sqrt{2} \cdot \sqrt{x}
\][/tex]
Since [tex]\(\sqrt{16} = 4\)[/tex]:
[tex]\[
\sqrt{16 \cdot 2 \cdot x} = 4 \cdot \sqrt{2x}
\][/tex]
Thus, [tex]\(c = 4\)[/tex].
3. Simplifying [tex]\(\sqrt{18 n}\)[/tex]:
[tex]\[
\sqrt{18 n} = \sqrt{9 \cdot 2 \cdot n}
\][/tex]
Using the same property:
[tex]\[
\sqrt{9 \cdot 2 \cdot n} = \sqrt{9} \cdot \sqrt{2} \cdot \sqrt{n}
\][/tex]
Since [tex]\(\sqrt{9} = 3\)[/tex]:
[tex]\[
\sqrt{9 \cdot 2 \cdot n} = 3 \sqrt{2n}
\][/tex]
Thus, [tex]\(e = 3\)[/tex].
4. Simplifying [tex]\(\sqrt{72 x^2}\)[/tex]:
[tex]\[
\sqrt{72 x^2} = \sqrt{36 \cdot 2 \cdot x^2}
\][/tex]
Again, applying the square root properties:
[tex]\[
\sqrt{36 \cdot 2 \cdot x^2} = \sqrt{36} \cdot \sqrt{2} \cdot \sqrt{x^2}
\][/tex]
Since [tex]\(\sqrt{36} = 6\)[/tex] and [tex]\(\sqrt{x^2} = x\)[/tex] for [tex]\(x \geq 0\)[/tex]:
[tex]\[
\sqrt{36 \cdot 2 \cdot x^2} = 6x \cdot \sqrt{2}
\][/tex]
Thus, [tex]\(g = 6\)[/tex].
Therefore, the values for the variables are:
[tex]\[
\begin{array}{l}
b = 2 \\
c = 4 \\
e = 3 \\
g = 6 \\
\end{array}
\][/tex]
