Simplifying $ \frac{3\sqrt{a^2}}{\sqrt{3}} \div 2a^{3/2}$

Question

I've been struggling for a while to answer this math question given to me, it reads:

Simplify this into both radical and exponential form: $$ \frac{3\sqrt{a^2}}{\sqrt{3}} \div 2a^{3/2}$$

I've tried to rationalize the denominator, though I am unsure if I should times $\dfrac{\sqrt{3}}{\sqrt{3}}$ to only $\dfrac{3\sqrt{a^2}}{\sqrt{3}}$ or the whole equation $\left(\dfrac{3 \sqrt{a^2}}{\sqrt{3}} \div 2 a^{3/2}\right)$.

So it's stumped me to figure out what to do now. Any help would be appreciated.

Answer 1

I believe I found an answer.

I started from exponential form and then converted it into radical form.

So I converted everything from

$\frac{3\sqrt{a^2}}{\sqrt{3}} \div 2a^{3/2}$

to

$\dfrac{3{a}^{\frac{2}{2}}}{{3}^{\frac{1}{2}}}\div2{a}^{\frac{3}{2}}$

Then I simplify even more by changing the division sign for $\div2{a}^{\frac{3}{2}}$ into $\times\dfrac{1}{2{a}^{-\frac{3}{2}}}$

Making this whole equation look like now

$ \dfrac{ 3 { a }^{ \frac{ 2 }{ 2 } } }{ { 3 }^{ \frac{ 1 }{ 2 } } } \times \dfrac{ 1 }{ 2 { a }^{ - \frac{ 3 }{ 2 } } } $

And then merge the two:

$\dfrac{ 3 { a }^{ \frac{ 2 }{ 2 } } \times 1 }{ { 3 }^{ \frac{ 1 }{ 2 } } \times 2 { a }^{ - \frac{ 3 }{ 2 } } }$

And since we shouldn't have the negative exponent in the denominator we move it up to the numerator changing its exponent from a negative to a positive. And will make

$ \dfrac{ 3 { a }^{ \frac{ 2 }{ 2 } } \cdot { a }^{ > \frac{ 3 }{ 2 } } }{ { 3 }^{ \frac{ 1 }{ 2 } } > \times 2 } $

into

$ \dfrac{ 3 { a }^{ \frac{ 5 }{ 2 } } }{ { 3 }^{ \frac{ > 1 }{ 2 } } \times 2 } $

And finally, we can express the 3 in the numerator as $ { 3 }^{ \frac{ 2 }{ 2 } }$, so

$ \dfrac{ { 3 }^{ \frac{ 2 }{ 2 } } { a }^{ \frac{ 5 }{ 2 } } }{ { 3 }^{ \frac{ 1 }{ 2 } } \times 2 } $ becomes $ \dfrac{ { 3 }^{ \frac{ 1 }{ 2 } } { a }^{ \frac{ 5 }{ 2 } } }{ 2 } $

Now the answer is in exponential form.

And to change it into radical form isn't too hard of a task either.

It is simply changing

$ \dfrac{ { 3 }^{ \frac{ 1 }{ 2 } } { a }^{ \frac{ 5 }{ 2 } } }{ 2 } $

Into

$ \dfrac{ \sqrt{ 3 \left( { a }^{ 5 } \right) \phantom{\tiny{!}}} }{ 2 } $

But, nonetheless, I think I've reached the answer. Thank you to everyone who guided me in this question.