Simplify (2×a/3)² : Squared Fraction Expression Solution

The task is to simplify $\left(\frac{2\times a}{3}\right)^2$.

First, applying the exponent rule for fractions, $\left(\frac{b}{c}\right)^n = \frac{b^n}{c^n}$, we have:

• $\left(\frac{2\times a}{3}\right)^2 = \frac{(2 \times a)^2}{3^2}$ - which represents performing the exponentiation separately on the numerator and denominator.

Now, simplify each part:

• The numerator: $(2 \times a)^2 = 2^2 \times a^2 = 4 \times a^2$.
• The denominator: $3^2 = 9$.

Thus, the expression simplifies to $\frac{4 \times a^2}{9}$.

We ensure the valid transformations based on the choices provided:

• Choice 1: $\frac{4\times a^2}{9}$, which matches what we calculated.
• Choice 2: $\frac{2^2\times a^2}{3^2}$, which is an equivalent form before final multiplication.
• Choice 3: $\frac{\left(2\times a\right)^2}{3^2}$, presenting the step before breaking down the $(2a)^2$.
• Choice 4: "All answers are correct", recognizing all transformations as valid.

Therefore, each choice represents correct steps or forms towards the simplified expression.

The correct answer is: All answers are correct.