The rational exponent form of a^{0.4} in radical form.

Change 0.4 into a fraction.

Multiply by $\frac{10}{10}$ to remove the decimal.

$a^{\frac{10\cdot 0.4}{10}}$

Multiply 10 by 0.4.

$a^{\frac{4}{10}}$

Cancel the common factor of 4 and 10

Factor 2 out of 4.

$a^{\frac{2(2)}{10}}$

Cancel the common factors.

Factor 2 out of 10.

$a^{\frac{2\cdot 2}{2\cdot 5}}$

Cancel the common factor.

$a^{\frac{2\cdot 2}{2\cdot 5}}$

Rewrite the expression.

$a^{\frac{2}{5}}$

Apply the rule $x^{\frac{m}{n}}=\sqrt[n]{x^m}$ to rewrite the exponentiation as a radical.

$\sqrt[5]{a^2}$