Question

what is the following product?
$sqrt3{4x^2} cdot sqrt3{4x^2}$
options:
$2left( sqrt3{4x^2}
ight)$
$4x^2$
$16x^4$
$sqrt3{16x^4}$

Explanation:

Step1: Recall the property of radicals

When multiplying two radicals with the same index, we can use the property $\sqrt[n]{a} \cdot \sqrt[n]{b}=\sqrt[n]{ab}$. Here, the index is 2 (square root), so $\sqrt{4x^2} \cdot \sqrt{4x^2}=\sqrt{(4x^2)(4x^2)}$.

Step2: Multiply the radicands

Multiply the terms inside the square root: $(4x^2)(4x^2) = 16x^4$. So now we have $\sqrt{16x^4}$.

Step3: Simplify the square root

We know that $\sqrt{16x^4}=\sqrt{(4x^2)^2}=4x^2$ (since the square root of a square is the absolute value, but assuming $x$ is such that $4x^2$ is non - negative, we can write it as $4x^2$). Alternatively, we can also think of it as $(\sqrt{4x^2})^2 = 4x^2$, or notice that $\sqrt{16x^4}=4x^2$ because $(4x^2)^2=16x^4$.

Answer:

$4x^2$ (corresponding to the option with $4x^2$)