Apply the Properties of Integer Exponents to generate equivalent expressions to ◂⋅▸3^7⋅3^−9 with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.
To apply the properties of integer exponents, we need to remember the following rules:
1. Product Rule: ${(a^m)(a^n) = a^{m+n}}$
2. Quotient Rule: $\displaystyle \left(\dfrac{a^m}{a^n}\right) = a^{m-n}$
3. Power Rule: ${(a^m)^n = a^{mn}}$
By applying the Product Rule, we can express the given expression as:
$\displaystyle 3^{7}\cdot 3^{ -9} = 3^{7-9} = 3^{-2}$
Since we want to have positive exponents only, we can apply the Quotient Rule:
$\displaystyle 3^{-2} = \dfrac{1}{3^{2}}$
Now, we can solve the expression:
$\displaystyle \dfrac{1}{3^{2}} = \dfrac{1}{3 \times 3} = \dfrac{1}{9}$
Therefore, the simplified fraction without remaining exponents is $\boxed{\dfrac{1}{9}}$.