To solve the problem of multiplying the fractions [tex]\(\frac{1}{9}\)[/tex] and [tex]\(\frac{3}{5}\)[/tex], let's go through the steps in detail:
1. Multiply the numerators:
[tex]\[
\frac{1}{9} \cdot \frac{3}{5} \Rightarrow 1 \cdot 3 = 3
\][/tex]
So, the numerator of the product is 3.
2. Multiply the denominators:
[tex]\[
\frac{1}{9} \cdot \frac{3}{5} \Rightarrow 9 \cdot 5 = 45
\][/tex]
So, the denominator of the product is 45.
Thus, the product of the fractions [tex]\(\frac{1}{9}\)[/tex] and [tex]\(\frac{3}{5}\)[/tex] is:
[tex]\[
\frac{3}{45}
\][/tex]
3. Simplify the fraction:
To reduce [tex]\(\frac{3}{45}\)[/tex] to its lowest terms, we need to find the greatest common divisor (GCD) of the numerator and the denominator.
The GCD of 3 and 45 is 3.
4. Divide the numerator and the denominator by the GCD:
[tex]\[
\frac{3 \div 3}{45 \div 3} = \frac{1}{15}
\][/tex]
So, [tex]\(\frac{3}{45}\)[/tex] simplifies to [tex]\(\frac{1}{15}\)[/tex].
Therefore, the product of [tex]\(\frac{1}{9}\)[/tex] and [tex]\(\frac{3}{5}\)[/tex], reduced to its lowest terms, is:
[tex]\[
\boxed{\frac{1}{15}}
\][/tex]
