To solve the equation [tex]\( 83,349 = 7^x \cdot 3^y \)[/tex], we need to determine the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] such that [tex]\( 7^x \)[/tex] and [tex]\( 3^y \)[/tex] are the prime factors of 83,349.
Let's begin by factorizing 83,349 into its prime components.
By examining the factors of 83,349, we find:
1. Prime factorization:
[tex]\[ 83,349 = 7^3 \cdot 3^5 \][/tex]
From this prime factorization, we can see that:
- The exponent [tex]\( x \)[/tex] corresponding to the prime factor 7 is [tex]\( 3 \)[/tex]
- The exponent [tex]\( y \)[/tex] corresponding to the prime factor 3 is [tex]\( 5 \)[/tex]
Now, we need to find the sum of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[
x + y = 3 + 5
\][/tex]
Finally, the sum of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] is:
[tex]\[
x + y = 8
\][/tex]
So, the value of [tex]\( x + y \)[/tex] is [tex]\( \boxed{8} \)[/tex].
