To solve the equation [tex]\(7^{-9} \cdot 7^{-3} = 7^x\)[/tex] for [tex]\(x\)[/tex], we need to utilize the properties of exponents.
Specifically, we use the property that states when you multiply two powers with the same base, you add the exponents. That is, [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex].
Here, the given equation is:
[tex]\[7^{-9} \cdot 7^{-3} = 7^x\][/tex]
We recognize that the base (7) is the same on both sides of the equation. This allows us to add the exponents on the left-hand side:
[tex]\[
-9 + (-3)
\][/tex]
Combining these, we get:
[tex]\[
-9 + (-3) = -12
\][/tex]
So, our equation simplifies to:
[tex]\[
7^{-12} = 7^x
\][/tex]
Since the bases are the same, we can set the exponents equal to each other:
[tex]\[
x = -12
\][/tex]
Thus, the solution to the equation [tex]\(7^{-9} \cdot 7^{-3} = 7^x\)[/tex] is:
[tex]\[
x = -12
\][/tex]
