Absolutely, let's go through each expression step-by-step to find the values of the variables [tex]\( b \)[/tex], [tex]\( c \)[/tex], [tex]\( e \)[/tex], and [tex]\( g \)[/tex].
1. Expression: [tex]\(\sqrt{50 x^2}\)[/tex]
[tex]\[
\sqrt{50 x^2} = \sqrt{25 \cdot 2 \cdot x^2}
\][/tex]
We recognize that [tex]\(25\)[/tex] is a perfect square, and we can separate it out:
[tex]\[
\sqrt{25 \cdot 2 \cdot x^2} = \sqrt{25} \cdot \sqrt{2} \cdot \sqrt{x^2} = 5 \cdot \sqrt{2} \cdot x
\][/tex]
Hence,
[tex]\[
\sqrt{50 x^2} = 5 x \sqrt{2}
\][/tex]
Therefore, [tex]\( b = 2 \)[/tex].
2. Expression: [tex]\(\sqrt{32 x}\)[/tex]
[tex]\[
\sqrt{32 x} = \sqrt{16 \cdot 2 \cdot x}
\][/tex]
We see that [tex]\(16\)[/tex] is a perfect square:
[tex]\[
\sqrt{16 \cdot 2 \cdot x} = \sqrt{16} \cdot \sqrt{2} \cdot \sqrt{x} = 4 \cdot \sqrt{2} \cdot \sqrt{x}
\][/tex]
Simplifying, we get:
[tex]\[
\sqrt{32 x} = 4 \sqrt{2 x}
\][/tex]
Therefore, [tex]\( c = 4 \)[/tex].
3. Expression: [tex]\(\sqrt{18 n}\)[/tex]
[tex]\[
\sqrt{18 n} = \sqrt{9 \cdot 2 \cdot n}
\][/tex]
Recognizing [tex]\(9\)[/tex] as a perfect square:
[tex]\[
\sqrt{9 \cdot 2 \cdot n} = \sqrt{9} \cdot \sqrt{2} \cdot \sqrt{n} = 3 \cdot \sqrt{2} \cdot \sqrt{n}
\][/tex]
Simplifying, we get:
[tex]\[
\sqrt{18 n} = 3 \sqrt{2 n}
\][/tex]
Therefore, [tex]\( e = 3 \)[/tex].
4. Expression: [tex]\(\sqrt{72 x^2}\)[/tex]
[tex]\[
\sqrt{72 x^2} = \sqrt{36 \cdot 2 \cdot x^2}
\][/tex]
Recognizing [tex]\(36\)[/tex] as a perfect square:
[tex]\[
\sqrt{36 \cdot 2 \cdot x^2} = \sqrt{36} \cdot \sqrt{2} \cdot \sqrt{x^2} = 6 \cdot \sqrt{2} \cdot x
\][/tex]
Simplifying, we get:
[tex]\[
\sqrt{72 x^2} = 6 x \sqrt{2}
\][/tex]
Therefore, [tex]\( g = 6 \)[/tex].
So, the values are:
[tex]\[
\begin{aligned}
b & = 2, \\
c & = 4, \\
e & = 3, \\
g & = 6.
\end{aligned}
\][/tex]
