Let's solve the equation [tex]\(\sqrt[3]{5} = 5^x\)[/tex].
First, recognize that [tex]\(\sqrt[3]{5}\)[/tex] is the cube root of 5. This can also be expressed as:
[tex]\[
5^{\frac{1}{3}}
\][/tex]
Thus, our equation becomes:
[tex]\[
5^{\frac{1}{3}} = 5^x
\][/tex]
Since the bases on both sides of the equation are the same (both are base 5), the exponents must be equal. Therefore, we can set the exponents equal to each other:
[tex]\[
\frac{1}{3} = x
\][/tex]
So, the value of [tex]\(x\)[/tex] is:
[tex]\[
x = \frac{1}{3}
\][/tex]
Expressing [tex]\(\frac{1}{3}\)[/tex] as a decimal:
[tex]\[
x = 0.3333333333333333
\][/tex]
Thus, the solution to the equation [tex]\(\sqrt[3]{5} = 5^x\)[/tex] is:
[tex]\[
x = 0.3333333333333333
\][/tex]
