Remember that using radical notation, fractionary exponents can be represented as follows:
[tex]a^{\frac{n}{m}}=\sqrt[m]{a^n}[/tex]
On the other hand, when the index of a radical does not appear, we understand that it is equal to 2:
[tex]\sqrt[]{a}=\sqrt[2]{a}[/tex]
Rewrite each of the radical factors of the given expression using fractionary exponents:
[tex]\sqrt[5]{x^3}\cdot\sqrt[]{x^4}=x^{\frac{3}{5}}\cdot x^{\frac{4}{2}}[/tex]
To simplify the expression, use the following rule of exponents:
[tex]a^n\times a^m=a^{n+m}[/tex]
Then:
[tex]x^{\frac{3}{5}}\cdot x^{\frac{4}{2}}=x^{\frac{3}{5}+\frac{4}{2}}[/tex]
To simplify the expression, solve the addition with fractions:
[tex]\frac{3}{5}+\frac{4}{2}[/tex]
Simplify the fraction 4/2 as 2/1:
[tex]\frac{3}{5}+\frac{4}{2}=\frac{3}{5}+\frac{2}{1}[/tex]
The leas common denominator for these fractions is 5. Rewrite 2/1 as 10/5 and add the fractions:
[tex]\frac{3}{5}+\frac{2}{1}=\frac{3}{5}+\frac{10}{5}=\frac{13}{5}[/tex]
Then:
[tex]x^{\frac{3}{5}+\frac{4}{2}}=x^{\frac{13}{5}}[/tex]
Therefore, the final expression in rational form, is:
[tex]\sqrt[5]{x^3}\cdot\sqrt[]{x^4}=x^{\frac{13}{5}}[/tex]
