Given that $\sqrt{a}\times a^3\div a^x = 1$, find the value of $x$. Simplify $4\div\left(\dfrac{x}{2}\right)^{-2}$, giving your answer in positive index.

1. Rewrite the square root as an index and combine the powers of $a$.

$\sqrt{a}\times a^3\div a^x=a^{\frac12}\times a^3\div a^x$

$=a^{\frac12+3-x}=a^{\frac72-x}$

2. Equate the index to zero because the value is 1.

Since $a^{\frac72-x}=1$, we need

$\frac72-x=0$

$x=\frac72$

1. Rewrite the negative index as a reciprocal with positive index.

$\left(\dfrac{x}{2}\right)^{-2}=\left(\dfrac{2}{x}\right)^2$

2. Divide by the fraction and simplify to a positive index expression.

$4\div\left(\dfrac{2}{x}\right)^2=4\div\dfrac{4}{x^2}=4\times\dfrac{x^2}{4}=x^2$

Alternatively, using the reciprocal form directly:
$4\div\left(\dfrac{x}{2}\right)^{-2}=4\div\dfrac{4}{x^2}=x^2$