It's better that you write \sqrt{z} instead, not sqrt[z]. If you wanted cube root though, you should write sqrt[3]{z}. The nth root is $\displaystyle \sqrt[n]{x}$ which is \sqrt[n]{x}.
Given: $\displaystyle 6.5 = \frac{52}{\sqrt{z}}$
Multiply by the multiplicative inverse of 6.5 on both sides: $\displaystyle 1 = \frac{52}{6.5\sqrt{z}}$
Multiply by $\displaystyle \sqrt{z}$ on both sides: $\displaystyle \sqrt{z} = \frac{52}{6.5}$
Square both sides (principal of powers): $\displaystyle z = \left(\frac{52}{6.5}\right)^2$