To determine which of the given options is equivalent to the expression [tex]\(\sqrt{7 x^2} + \sqrt{3 x}\)[/tex], let's examine each option one by one.
Given expression:
[tex]\[ \sqrt{7 x^2} + \sqrt{3 x} \][/tex]
1. Option A: [tex]\(\sqrt{\frac{2 x}{3}}\)[/tex]
Simplify [tex]\(\sqrt{\frac{2 x}{3}}\)[/tex] to compare with the given expression. This expression, as it stands, does not resemble either term in the given expression. It also won't simplify directly to [tex]\(\sqrt{7 x^2} + \sqrt{3 x}\)[/tex]. Hence, this option does not look equivalent.
2. Option B: [tex]\(\sqrt{21 x^2}\)[/tex]
Simplify [tex]\(\sqrt{21 x^2}\)[/tex]:
[tex]\[ \sqrt{21 x^2} = \sqrt{21} \cdot \sqrt{x^2} = \sqrt{21} \cdot |x| = \sqrt{21} x \][/tex]
This does not match [tex]\(\sqrt{7 x^2} + \sqrt{3 x}\)[/tex] either in form. Therefore, this option does not look equivalent.
3. Option C: [tex]\(\sqrt{\frac{7 x^3}{3}}\)[/tex]
Simplify [tex]\(\sqrt{\frac{7 x^3}{3}}\)[/tex]:
[tex]\[ \sqrt{\frac{7 x^3}{3}} = \sqrt{7 x^3} \cdot \frac{1}{\sqrt{3}} = \frac{\sqrt{7 x^3}}{\sqrt{3}} \][/tex]
This simplification does not form an expression similar to [tex]\(\sqrt{7 x^2} + \sqrt{3 x}\)[/tex], hence this option does not look equivalent either.
4. Option D: [tex]\(x \sqrt{\frac{7 x}{3}}\)[/tex]
Simplify [tex]\(x \sqrt{\frac{7 x}{3}}\)[/tex]:
[tex]\[ x \sqrt{\frac{7 x}{3}} = x \cdot \sqrt{7 x} \cdot \frac{1}{\sqrt{3}} = x \cdot \sqrt{7} \cdot \sqrt{x} \cdot \frac{1}{\sqrt{3}} = x \cdot \sqrt{\frac{7 x}{3}} \][/tex]
Again, this does not simplify to [tex]\(\sqrt{7 x^2} + \sqrt{3 x}\)[/tex] and does not seem equivalent.
Based on examining each option, none of the choices simplify to match the expression [tex]\(\sqrt{7 x^2} + \sqrt{3 x}\)[/tex] exactly. Therefore, we conclude:
None of the given options are equivalent to the expression [tex]\(\sqrt{7 x^2} + \sqrt{3 x}\)[/tex].
