Answer
${\bf 0.766}\;R$
Work Step by Step
We know that the magnitude of the magnetic field at the center of the axis of a current loop is given by
$$B=\dfrac{\mu_0 I R^2}{2(z^2+R^2)^{3/2}}\tag 1$$
At the center of the loop $z=0$, so
$$B_0=\dfrac{\mu_0 I R^2}{2R^3}\tag 2$$
Now we need to find the distance on the axis of a current loop when the magnetic field is half $B_0$,
$$B=\dfrac{B_0}{2}$$
Plugging from (2),
$$B=\dfrac{\mu_0 I R^2}{4R^3}$$
Using (1),
$$\dfrac{\mu_0 I R^2}{2(z^2+R^2)^{3/2}}=\dfrac{\mu_0 I R^2}{4R^3}$$
Hence,
$$ (z^2+R^2)^{3/2}=2R^3$$
$$ z^2+R^2 =(2R^3)^{2/3}$$
$$ z^2 =(2R^3)^{2/3}-R^2$$
$$ z^2 = 2^{2/3}R^2-R^2=R^2(2^{2/3}-1)$$
Thus,
$$\boxed{z=\color{red}{\bf 0.766}\;R}$$
