Sure! Let's go through the problem step-by-step to find the product of the given fractions and simplify it.
We have the fractions [tex]\(\frac{6}{13}\)[/tex], [tex]\(\frac{9}{77}\)[/tex], and [tex]\(\frac{11}{2}\)[/tex]. Our goal is to multiply these together and simplify the result.
Step 1: Multiply the numerators and the denominators separately.
The numerators are [tex]\(6 \times 9 \times 11\)[/tex]:
[tex]\[
6 \times 9 \times 11 = 594
\][/tex]
The denominators are [tex]\(13 \times 77 \times 2\)[/tex]:
[tex]\[
13 \times 77 \times 2 = 2002
\][/tex]
So, the product of the fractions is:
[tex]\[
\frac{6}{13} \times \frac{9}{77} \times \frac{11}{2} = \frac{594}{2002}
\][/tex]
Step 2: Simplify the fraction [tex]\(\frac{594}{2002}\)[/tex] to its lowest terms.
To do this, we find the greatest common divisor (GCD) of the numerator and the denominator and divide both by this GCD.
The GCD of 594 and 2002 is 22 (this can be determined through the Euclidean algorithm or using a suitable method for finding GCD).
Step 3: Simplify by dividing both the numerator and the denominator by the GCD:
[tex]\[
\frac{594 \div 22}{2002 \div 22} = \frac{27}{91}
\][/tex]
So, the simplified answer to [tex]\(\frac{6}{13} \times \frac{9}{77} \times \frac{11}{2}\)[/tex] is:
[tex]\[
\boxed{\frac{27}{91}}
\][/tex]
