Question

simplify using the quotient property


5^2
------
^5√25

Answer 1

To simplify the expression \(\frac{5^2}{\sqrt[5]{25}}\), we can start by rewriting \(25\) in terms of \(5\):

\[
25 = 5^2
\]

Thus, we can write the fifth root of \(25\) as:

\[
\sqrt[5]{25} = \sqrt[5]{5^2} = 5^{\frac{2}{5}}
\]

Now, substituting this back into the original expression, we have:

\[
\frac{5^2}{\sqrt[5]{25}} = \frac{5^2}{5^{\frac{2}{5}}}
\]

Using the quotient property of exponents, which states that \(\frac{a^m}{a^n} = a^{m-n}\), we can simplify:

\[
\frac{5^2}{5^{\frac{2}{5}}} = 5^{2 - \frac{2}{5}}
\]

To perform the subtraction \(2 - \frac{2}{5}\), we first need to express \(2\) as a fraction with a common denominator:

\[
2 = \frac{10}{5}
\]

Now we can subtract:

\[
2 - \frac{2}{5} = \frac{10}{5} - \frac{2}{5} = \frac{10 - 2}{5} = \frac{8}{5}
\]

Therefore:

\[
5^{2 - \frac{2}{5}} = 5^{\frac{8}{5}}
\]

The final simplified expression is:

\[
\frac{5^2}{\sqrt[5]{25}} = 5^{\frac{8}{5}}
\]