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How would you go about simplifying the expression $\frac{75}{8}\times\sqrt{\frac{a^3}{9}-\frac{a^3}{25}}$?

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    find the common denominator...and add the terms under the radical togheter. separate what is under the radical into factor that are perfect squares and those that are not. Those that are, can be brought outside. – 2017-02-28

2 Answers 2

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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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    Well put (+1) $\color{white}{\text{plenty of whitespace}}$ – 2017-03-02
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    LOL @SimplyBeautifulArt I see you so often :P – 2017-03-02
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    Well, there is a reason...[I'm active a lot](https://i.stack.imgur.com/WOOY1.png) – 2017-03-02
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    @SimplyBeautifulArt How did you put that whitespace? Another suggestion is `${}{}{}{}{}$` – 2018-03-17
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    No, hes more clever then that, he did this – 2018-03-17
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    \color{white}{\text{plenty of whitespace}} .... which if you were to surround in dollars would be ~hello~$\color{white}{\text{plenty of whitespace}}$ – 2018-03-17
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    @idk nice question – 2018-03-17
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    >_> giving out my secrets lol – 2018-03-19
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    @SimplyBeautifulArt neat trick, I actually highlighted the entire thing and saw that haha – 2018-03-19
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$$\frac{75}{8}\sqrt{\frac{a^3}{9}-\frac{a^3}{25}}=\frac{75}{8}\sqrt{\frac{25a^3-9a^3}{225}}=\frac{75}{8}\frac{\sqrt{16}\sqrt{a^3}}{\sqrt{225}}=\frac{75}{8}\frac{4}{15}\sqrt{a^3}=2.5\sqrt{a^3}$$

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    Why didn't you factor out $a^2$ from the radical sign at the same time you factored out $16$ and $225$? $\sqrt {a^3 } = a\sqrt{a}$. – 2017-03-01
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    @fleablood $a\sqrt{a}$ is not more simple then $\sqrt{a^3}$ – 2017-03-01