The fractional change in the magnetic field intensity at a distance...

As we know,

${B}_{axis} = \dfrac{{\mu}_{0}{iR}^{2}}{2{\left({R}^{2} + {x}^{2}\right)}^{3 / 2}}\text{where}, R = a$

${B}_{\text{centre}} = \dfrac{{\mu}_{0}i}{2a}$

$∴ {B}_{\text{axis}} = \dfrac{{\mu}_{0}{ia}^{2}}{2{\left({a}^{2} + {r}^{2}\right)}^{3 / 2}}$

$∴$ fractional change in magnetic field $= \dfrac{\dfrac{{\mu}_{0}i}{2a} - \dfrac{{\mu}_{0}i{a}^{2}}{2{\left({a}^{2} + {r}^{2}\right)}^{3 / 2}}}{\dfrac{{\mu}_{0}i}{2a}} = 1 - \dfrac{1}{{\left[1 + \left(\dfrac{{r}^{2}}{{a}^{2}}\right)\right]}^{3 / 2}}$

$\approx 1 - \left[1 - \dfrac{3}{2}\dfrac{{r}^{2}}{{a}^{2}}\right] = \dfrac{3}{2}\dfrac{{r}^{2}}{{a}^{2}}$