#### Answer

$-\dfrac{8}{13}$ and $\dfrac{8}{13}$

#### Work Step by Step

Note that
$\left(\frac{8}{13}\right)^2=\frac{64}{169}$, so $\frac{8}{13}$ is a real square root of $\frac{64}{169}$, and
$\left(-\frac{8}{13}\right)^2=\frac{64}{169}$, so $-\frac{8}{13}$ is a also real square root of $\frac{64}{169}$.
Thus, the real square roots of $\frac{64}{169}$ are $-\frac{8}{13}$ and $\frac{8}{13}$.