<s>[INST] How to simplify $\frac{75}{8}\times\sqrt{\frac{a^3}{9}-\frac{a^3}{25}}$ How would you go about simplifying the expression $\frac{75}{8}\times\sqrt{\frac{a^3}{9}-\frac{a^3}{25}}$?
 [/INST] We have:
$$\frac{75}{8}\times\sqrt{\frac{a^3}{9}-\frac{a^3}{25}}$$
Let's deal with the stuff inside the square root first.
$${\frac{a^3}{9}-\frac{a^3}{25}}$$
Finding a common denominator and simplifying:
$${\frac{(a^3)(25)-(a^3)(9)}{225}}$$
$${\frac{25a^3-9a^3}{225}}$$
$${\frac{16a^3}{225}}$$
$${\frac{16}{225}}\times a^3$$
We can take square root of these things.
$$\sqrt{\frac{16}{225}}= \frac{\sqrt{16}}{\sqrt{225}}=\frac{4}{15}$$
$$\sqrt{a^3}=(a^3)^{(1/2)}=a^{(3/2)}$$
So, 
$$\sqrt{\frac{a^3}{9}-\frac{a^3}{25}} = \frac{4a^{3/2}}{15}$$
Now multiply by $\frac{75}{8}$.
$$\frac{75}{8}\times\frac{4a^{3/2}}{15} = \frac{5}{2}\times\frac{1a^{3/2}}{1} = \frac{5a^{3/2}}{2}$$
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