To find the value of [tex]\(\frac{x}{y}\)[/tex] given [tex]\(x = 3^m\)[/tex] and [tex]\(y = 3^{m+2}\)[/tex], we proceed with the following steps:
1. Substitute the expressions for [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
[tex]\[
x = 3^m
\][/tex]
[tex]\[
y = 3^{m+2}
\][/tex]
2. Set up the fraction [tex]\(\frac{x}{y}\)[/tex]:
[tex]\[
\frac{x}{y} = \frac{3^m}{3^{m+2}}
\][/tex]
3. Apply the properties of exponents:
One of the properties of exponents is [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex]. Using this property, we can simplify the fraction:
[tex]\[
\frac{3^m}{3^{m+2}} = 3^{m - (m+2)}
\][/tex]
4. Simplify the exponent:
[tex]\[
3^{m - (m+2)} = 3^{m - m - 2} = 3^{-2}
\][/tex]
5. Rewrite the negative exponent as a positive exponent:
[tex]\[
3^{-2} = \frac{1}{3^2}
\][/tex]
6. Calculate the final numerical value:
[tex]\[
3^2 = 9
\][/tex]
Therefore:
[tex]\[
\frac{1}{3^2} = \frac{1}{9}
\][/tex]
Thus, the value of [tex]\(\frac{x}{y}\)[/tex] is:
[tex]\[
\frac{x}{y} = \frac{1}{9}
\][/tex]
