Certainly! To multiply two radical expressions, we'll need to use the properties of exponents and radicals.
Given the expressions:
[tex]\[ \sqrt[5]{x} \cdot \sqrt[8]{x} \][/tex]
Let's express the radicals in exponential form. Recall that [tex]\(\sqrt[n]{x} = x^{\frac{1}{n}}\)[/tex].
Therefore,
[tex]\[ \sqrt[5]{x} = x^{\frac{1}{5}} \][/tex]
[tex]\[ \sqrt[8]{x} = x^{\frac{1}{8}} \][/tex]
Now, we'll multiply these exponential expressions. When you multiply expressions that have the same base, you add the exponents:
[tex]\[ x^{\frac{1}{5}} \cdot x^{\frac{1}{8}} = x^{\frac{1}{5} + \frac{1}{8}} \][/tex]
Next, we need to add the fractions [tex]\(\frac{1}{5}\)[/tex] and [tex]\(\frac{1}{8}\)[/tex]. To do this, we'll find a common denominator. The least common multiple of 5 and 8 is 40. Let's convert the fractions:
[tex]\[ \frac{1}{5} = \frac{8}{40} \][/tex]
[tex]\[ \frac{1}{8} = \frac{5}{40} \][/tex]
Now, add these fractions together:
[tex]\[ \frac{8}{40} + \frac{5}{40} = \frac{13}{40} \][/tex]
So we have:
[tex]\[ x^{\frac{1}{5} + \frac{1}{8}} = x^{\frac{13}{40}} \][/tex]
To express this result in radical form, recall that [tex]\( x^{\frac{m}{n}} = \sqrt[n]{x^m} \)[/tex]. Therefore,
[tex]\[ x^{\frac{13}{40}} = \sqrt[40]{x^{13}} \][/tex]
Thus, the answer to multiplying [tex]\(\sqrt[5]{x} \cdot \sqrt[8]{x}\)[/tex] and leaving it in radical form is:
[tex]\[ \sqrt[40]{x^{13}} \][/tex]
