To simplify: The given expression (2a)(3{a2})

Step 1
Law of exponents:
For any rational exponent ${\displaystyle \frac{m}{n}}$ in lowest terms, where m and n are integers and ${\displaystyle n>0,}$ we define
${\displaystyle a^{\frac{m}{n}}={\left(\sqrt{n}\left\{a\right\}\right)}^m=\sqrt{n}\left\{a^m\right\}}$
If n is even, then we require that ${\displaystyle a\ge 0}$
Step 2
Consider the given expression,
${\displaystyle \left(2\sqrt{a}\right)\left(\sqrt{3}\left\{a^2\right\}\right)}$
By using the law of exponents,
${\displaystyle \left(2\sqrt{a}\right)\left\{\sqrt{3}\left\{a^2\right\}\right)=\left(2a^{\frac{1}{2}}\right)\left(a^{\frac{2}{3}}\right)}$
${\displaystyle =2a^{\frac{1}{2}}\cdot a^{\frac{2}{3}}}$
Applying the exponent rule: ${\displaystyle a^b\cdot a^c=a^{b+c},}$ we get
${\displaystyle =2a^{\frac{1}{2}+\frac{2}{3}}}$
${\displaystyle =2a^{\frac{7}{6}}}$
Final Statement:
The simplified form of ${\displaystyle \left(2\sqrt{a}\right)\left(\sqrt{3}\left\{a^2\right\}\right)}$ is ${\displaystyle 2a^{\frac{7}{6}}}$