To evaluate the expression [tex]\(\left(\frac{1}{5}\right)^4\)[/tex], we follow these steps:
1. Identify the base and the exponent in the expression. Here, the base is [tex]\(\frac{1}{5}\)[/tex] and the exponent is 4.
2. Understand that raising a fraction to an exponent means multiplying the fraction by itself as many times as indicated by the exponent. In this case, we will multiply [tex]\(\frac{1}{5}\)[/tex] by itself 4 times.
[tex]\[
\left(\frac{1}{5}\right)^4 = \frac{1}{5} \times \frac{1}{5} \times \frac{1}{5} \times \frac{1}{5}
\][/tex]
3. Calculate the multiplication of the fractions manually or recognize that each multiplication of [tex]\(\frac{1}{5}\)[/tex] results in the base raised to the current power step.
4. Understanding that [tex]\(\left(a^m \times a^n = a^{m+n}\right)\)[/tex], we can simplify the computations.
5. So, the calculation proceeds as follows:
[tex]\[
\frac{1}{5} \times \frac{1}{5} = \frac{1}{25}
\][/tex]
Next,
[tex]\[
\frac{1}{25} \times \frac{1}{5} = \frac{1}{125}
\][/tex]
Finally,
[tex]\[
\frac{1}{125} \times \frac{1}{5} = \frac{1}{625}
\][/tex]
6. Thus,
[tex]\[
\left(\frac{1}{5}\right)^4 = \frac{1}{625}
\][/tex]
7. Converting [tex]\(\frac{1}{625}\)[/tex] to its decimal form, we find the answer to be approximately [tex]\(0.0016000000000000003\)[/tex].
So, the final result is:
[tex]\[
\left(\frac{1}{5}\right)^4 = 0.0016000000000000003
\][/tex]
