Sure, let's fill in the values for the variables step-by-step.
### Step-by-Step Solutions:
1. Expression: [tex]\(\sqrt{50 x^2}=\sqrt{25 \cdot 2 \cdot x^2}=5 x \sqrt{b}\)[/tex]
- Given [tex]\(\sqrt{50 x^2}\)[/tex], we can factor 50 as [tex]\(25 \cdot 2\)[/tex] and recognize that [tex]\(x^2\)[/tex] is a perfect square.
- Thus, [tex]\(\sqrt{50 x^2} = \sqrt{25 \cdot 2 \cdot x^2} = \sqrt{25} \cdot \sqrt{2} \cdot \sqrt{x^2} = 5 x \sqrt{2}\)[/tex].
Therefore, [tex]\(b = 2\)[/tex].
2. Expression: [tex]\(\sqrt{32 x}=\sqrt{16 \cdot 2 \cdot x} = c \sqrt{2 x}\)[/tex]
- For [tex]\(\sqrt{32 x}\)[/tex], we factor 32 as [tex]\(16 \cdot 2\)[/tex].
- So, [tex]\(\sqrt{32 x} = \sqrt{16 \cdot 2 \cdot x} = \sqrt{16} \cdot \sqrt{2 x} = 4 \sqrt{2 x}\)[/tex].
Therefore, [tex]\(c = 4\)[/tex].
3. Expression: [tex]\(\sqrt{18 n}=\sqrt{9 \cdot 2 \cdot n}=e \sqrt{2 n}\)[/tex]
- For [tex]\(\sqrt{18 n}\)[/tex], we factor 18 as [tex]\(9 \cdot 2\)[/tex].
- Thus, [tex]\(\sqrt{18 n} = \sqrt{9 \cdot 2 \cdot n} = \sqrt{9} \cdot \sqrt{2 n} = 3 \cdot \sqrt{2 n}\)[/tex].
Therefore, [tex]\(e = 3\)[/tex].
4. Expression: [tex]\(\sqrt{72 x^2}=\sqrt{36 \cdot 2 \cdot x^2}=g x \sqrt{2}\)[/tex]
- For [tex]\(\sqrt{72 x^2}\)[/tex], we factor 72 as [tex]\(36 \cdot 2\)[/tex] and recognize that [tex]\(x^2\)[/tex] is a perfect square.
- Thus, [tex]\(\sqrt{72 x^2} = \sqrt{36 \cdot 2 \cdot x^2} = \sqrt{36} \cdot \sqrt{2} \cdot \sqrt{x^2} = 6 x \sqrt{2}\)[/tex].
Therefore, [tex]\(g = 6\)[/tex].
### Summary:
[tex]\[
\begin{array}{l}
\sqrt{50 x^2} = 5 x \sqrt{2}, \quad b = 2 \\
\sqrt{32 x} = 4 \sqrt{2 x}, \quad c = 4 \\
\sqrt{18 n} = 3 \sqrt{2 n}, \quad e = 3 \\
\sqrt{72 x^2} = 6 x \sqrt{2}, \quad g = 6 \\
\end{array}
\][/tex]
Thus, the values for [tex]\(b\)[/tex], [tex]\(c\)[/tex], [tex]\(e\)[/tex], and [tex]\(g\)[/tex] are 2, 4, 3, and 6 respectively.
