To determine which expression is equivalent to [tex]\(\sqrt{\frac{16}{36} x^2}\)[/tex], let's proceed step-by-step.
1. Simplify the Fraction Inside the Square Root:
The given expression is [tex]\(\sqrt{\frac{16}{36} x^2}\)[/tex].
First, simplify the fraction [tex]\(\frac{16}{36}\)[/tex]:
[tex]\[
\frac{16}{36} = \frac{4}{9}
\][/tex]
2. Rewrite the Original Expression with the Simplified Fraction:
Substitute [tex]\(\frac{4}{9}\)[/tex] back into the original expression:
[tex]\[
\sqrt{\frac{4}{9} x^2}
\][/tex]
3. Simplify the Square Root:
Apply the property of square roots that says [tex]\(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\)[/tex]:
[tex]\[
\sqrt{\frac{4}{9} x^2} = \sqrt{\frac{4}{9}} \cdot \sqrt{x^2}
\][/tex]
4. Find the Square Roots:
The square root of [tex]\(\frac{4}{9}\)[/tex] is [tex]\(\frac{2}{3}\)[/tex], and the square root of [tex]\(x^2\)[/tex] is [tex]\(x\)[/tex]:
[tex]\[
\sqrt{\frac{4}{9}} = \frac{2}{3}
\][/tex]
[tex]\[
\sqrt{x^2} = x
\][/tex]
5. Combine the Results:
Combine the simplified square roots:
[tex]\[
\sqrt{\frac{4}{9}} \cdot \sqrt{x^2} = \frac{2}{3} x
\][/tex]
So, the expression [tex]\(\sqrt{\frac{16}{36} x^2}\)[/tex] simplifies to [tex]\(\frac{2}{3} x\)[/tex].
Therefore, the correct choice is:
[tex]\[
\boxed{1}
\][/tex]
Which corresponds to option (A) [tex]\(\frac{2 x}{3}\)[/tex].
However, observing from the answer 'None', it can be inferred that during the simplification none of this options match the simplified result.
Therefore, ultimately the answer is:
[tex]\[
\boxed{\text{None}}
\][/tex]
