To simplify the expression [tex]\(\frac{9^{\frac{3}{5}}}{9^{\frac{1}{5}}}\)[/tex], we can apply the Quotient Property of exponents. The Quotient Property states that when you divide like bases, you subtract the exponents: [tex]\(x^a / x^b = x^{a-b}\)[/tex].
Given the expression:
[tex]\[ \frac{9^{\frac{3}{5}}}{9^{\frac{1}{5}}}, \][/tex]
we identify the base as 9 and the exponents as [tex]\(\frac{3}{5}\)[/tex] and [tex]\(\frac{1}{5}\)[/tex].
According to the Quotient Property, we subtract the exponent in the denominator from the exponent in the numerator:
[tex]\[ \frac{9^{\frac{3}{5}}}{9^{\frac{1}{5}}} = 9^{(\frac{3}{5} - \frac{1}{5})}. \][/tex]
Now, subtract the exponents:
[tex]\[ \frac{3}{5} - \frac{1}{5} = \frac{3-1}{5} = \frac{2}{5}. \][/tex]
Therefore, the simplified exponent in the equivalent expression is:
[tex]\[ 9^{\frac{2}{5}}. \][/tex]
The simplified exponent is [tex]\(\frac{2}{5}\)[/tex].
