Question

1. \\(\dfrac{5^{1/5}}{5}\\)

Explanation:

Step1: Use exponent rule \( \frac{a^m}{a^n}=a^{m - n} \)

We can rewrite the expression \( \frac{5^{\frac{1}{5}}}{5} \) as \( \frac{5^{\frac{1}{5}}}{5^1} \). By the rule of exponents for division, when we divide two numbers with the same base, we subtract the exponents. So this becomes \( 5^{\frac{1}{5}-1} \).

Step2: Simplify the exponent

Calculate \( \frac{1}{5}-1 \). We can write \( 1 \) as \( \frac{5}{5} \), so \( \frac{1}{5}-\frac{5}{5}=\frac{1 - 5}{5}=\frac{-4}{5} \). So the expression is now \( 5^{-\frac{4}{5}} \). Alternatively, we can also use the rule \( a^{-n}=\frac{1}{a^n} \), so \( 5^{-\frac{4}{5}}=\frac{1}{5^{\frac{4}{5}}} \). But if we want to simplify it in terms of positive exponents or a radical, \( 5^{\frac{4}{5}}=\sqrt[5]{5^4}=\sqrt[5]{625} \), so the expression is \( \frac{1}{\sqrt[5]{625}} \) or \( 5^{-\frac{4}{5}} \). However, if we just simplify the exponent subtraction step:
\( \frac{5^{\frac{1}{5}}}{5}=5^{\frac{1}{5}-1}=5^{-\frac{4}{5}}=\frac{1}{5^{\frac{4}{5}}}=\frac{1}{\sqrt[5]{5^4}}=\frac{1}{\sqrt[5]{625}} \)
But maybe a simpler way is to recognize that \( 5 = 5^1 \), so \( \frac{5^{\frac{1}{5}}}{5^1}=5^{\frac{1}{5}-1}=5^{-\frac{4}{5}} \), or if we want to write it as a single fraction exponent, or we can also calculate the numerical value. Let's compute the numerical value:
First, \( 5^{\frac{1}{5}}=\sqrt[5]{5}\approx1.3797 \)
Then divide by \( 5 \): \( \frac{1.3797}{5}\approx0.2759 \)
Or using the exponent rule:
\( 5^{-\frac{4}{5}}=\frac{1}{5^{\frac{4}{5}}}=\frac{1}{\sqrt[5]{5^4}}=\frac{1}{\sqrt[5]{625}}\approx\frac{1}{3.6239}\approx0.2759 \)

Answer:

The simplified form is \( 5^{-\frac{4}{5}} \) (or \( \frac{1}{\sqrt[5]{625}} \) or approximately \( 0.276 \))