You can distribute square roots between numerator and denominator:
[tex] \sqrt{\dfrac{64}{9}} = \dfrac{\sqrt{64}}{\sqrt{9}} [/tex]
Now, the square root of a number [tex] y [/tex] is a number [tex] x [/tex] such that [tex] x^2 = y [/tex]
So, [tex] \sqrt{64} [/tex] is a number that, when squared, gives 64. This number is 8, since [tex] 8^2 = 64 [/tex]
For the same reason, [tex] \sqrt{9}=3[/tex]
So, you have
[tex] \sqrt{\dfrac{64}{9}} = \dfrac{\sqrt{64}}{\sqrt{9}} = \dfrac{8}{3}[/tex]
