To evaluate the expression [tex]\(\left(\frac{1}{5}\right)^4\)[/tex], let's follow these steps closely:
1. Understand the base and the exponent: Here, [tex]\(\frac{1}{5}\)[/tex] is the base, and [tex]\(4\)[/tex] is the exponent.
2. Interpretation of exponentiation: Raising a fraction to an exponent means multiplying that fraction by itself as many times as indicated by the exponent. So, we have:
[tex]\[
\left(\frac{1}{5}\right)^4 = \frac{1}{5} \times \frac{1}{5} \times \frac{1}{5} \times \frac{1}{5}
\][/tex]
3. Multiply the fractions: To multiply fractions, you multiply the numerators together and the denominators together.
The numerator is:
[tex]\[
1 \times 1 \times 1 \times 1 = 1
\][/tex]
The denominator is:
[tex]\[
5 \times 5 \times 5 \times 5
\][/tex]
4. Simplify the denominator:
[tex]\[
5 \times 5 = 25
\][/tex]
[tex]\[
25 \times 5 = 125
\][/tex]
[tex]\[
125 \times 5 = 625
\][/tex]
Therefore, the denominator is 625.
5. Combine the results: Combining the calculated numerator and denominator gives us:
[tex]\[
\frac{1}{625}
\][/tex]
6. Convert to decimal form: To express [tex]\(\frac{1}{625}\)[/tex] in decimal form, we need to perform the division [tex]\( \frac{1}{625} \)[/tex].
7. Decimal fraction outcome: Performing the division, we find that:
[tex]\[
\frac{1}{625} = 0.0016000000000000003
\][/tex]
Thus,
[tex]\[
\left(\frac{1}{5}\right)^4 = 0.0016000000000000003
\][/tex]
