$L[t*\sqrt{t}] = L[t*t^{1/2}] = L[t^{3/2}]$
$L[t^{n}] =\frac{\Gamma n}{s^{n+1}}$ so $L[t^{3/2}]= \frac{\Gamma \frac {3}{2}}{s^\frac {5}{2}}$
$ = \frac{\frac {1}{2} \Gamma \frac {1}{2}}{s^\frac {5}{2}}$
$ = \frac {\sqrt{\pi}} {2 s^\frac {5}{2}}$
but my texbook and this answer says that the correct answer is $\frac{1}{s^{5/2}} \cdot \frac{3\sqrt{\pi}}{4}$
where did I make a mistake?
Actually, in Laplace transformation, $$ F(p)={\mathcal{L}}\left\{ x^n:p \right\}=\dfrac{\Gamma(n+1)}{p^{n+1}}~~~~~~~~~~~~\text{for}~~~p >-1$$ and $$ F(p)={\mathcal{L}}\left\{ x^n:p \right\}=\dfrac{n!}{p^{n+1}}~~~~~~~~~~~~\text{for}~~~p =0,~1,~2,~\cdots$$ For your case $~n=3/2~,$ so $$L[t^{3/2}]= \frac{\Gamma \left(\frac {3}{2}+1\right)}{s^\frac {5}{2}}=\frac{\frac {3}{2}\cdot\Gamma \left(\frac {3}{2}\right)}{s^\frac {5}{2}}=\frac{\frac {3}{2}\cdot\frac {1}{2}\cdot\Gamma \left(\frac {1}{2}\right)}{s^\frac {5}{2}}=\frac{\frac {3}{4}\cdot\sqrt \pi}{s^\frac {5}{2}}=\dfrac{3\sqrt \pi}{4}s^{-5/2}~.$$ as $~ Γ(n+1)=nΓ(n)~$ and $~Γ(1/2)=\sqrt \pi~.$