To solve this problem, we need to apply the properties of integer exponents. The property we will use here is the product of powers property, which states that when multiplying exponential expressions with the same base, we can add their exponents.
The given expression is:
3^7 ⋅ 3^(-9)
To generate an equivalent expression with only positive exponents, we can rewrite 3^(-9) as 1/3^9.
Now we have:
3^7 ⋅ 1/3^9
Applying the product of powers property, we add the exponents:
3^(7-9), which simplifies to 3^(-2).
To solve the expression, we need to write 3^(-2) as a fraction with no remaining exponents. To convert a negative exponent to a positive exponent, we can write it as the reciprocal of the base raised to the positive exponent. So we have:
3^(-2) = 1 / 3^2
Simplifying 3^2, we get 9.
Therefore, 3^7 ⋅ 3^(-9) is equivalent to 1/9.