To multiply the given radicals [tex]\(\sqrt[5]{x}\)[/tex] and [tex]\(\sqrt[8]{x}\)[/tex], follow these steps:
1. Express each radical using exponents:
The expression [tex]\(\sqrt[5]{x}\)[/tex] can be rewritten as [tex]\(x^{\frac{1}{5}}\)[/tex], and [tex]\(\sqrt[8]{x}\)[/tex] can be rewritten as [tex]\(x^{\frac{1}{8}}\)[/tex].
2. Multiply the expressions using the property of exponents:
When you multiply [tex]\(x^{\frac{1}{5}}\)[/tex] and [tex]\(x^{\frac{1}{8}}\)[/tex], you add the exponents:
[tex]\[
x^{\frac{1}{5}} \cdot x^{\frac{1}{8}} = x^{\left(\frac{1}{5} + \frac{1}{8}\right)}
\][/tex]
3. Calculate the sum of the exponents:
Find the common denominator to add the fractions. Here, both [tex]\(1/5\)[/tex] and [tex]\(1/8\)[/tex] can be added directly as decimals:
[tex]\[
\frac{1}{5} + \frac{1}{8} = 0.2 + 0.125 = 0.325
\][/tex]
4. Express the result:
Therefore, the result is:
[tex]\[
x^{0.325}
\][/tex]
This can be converted back to radical form, although for most practical purposes in this context, it is often left in exponent form [tex]\(x^{0.325}\)[/tex].
Thus, the product of [tex]\(\sqrt[5]{x}\)[/tex] and [tex]\(\sqrt[8]{x}\)[/tex] is [tex]\( x^{0.325} \)[/tex].
