Apply the Properties of Integer Exponents to generate equivalent expressions to 3^7⋅3^−9

Question

with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.(1 point)
3^7⋅3^−9 = BLANK

Answer 1

To simplify the expression \( 3^7 \cdot 3^{-9} \) using the properties of integer exponents, we can apply the property that states:

\[ a^m \cdot a^n = a^{m+n} \]

In this case, \(m = 7\) and \(n = -9\). Thus, we can combine the exponents:

\[ 3^7 \cdot 3^{-9} = 3^{7 + (-9)} = 3^{7 - 9} = 3^{-2} \]

Next, we rewrite \(3^{-2}\) with only positive exponents. The property for negative exponents states:

\[ a^{-n} = \frac{1}{a^n} \]

So, we have:

\[ 3^{-2} = \frac{1}{3^{2}} = \frac{1}{9} \]

Thus, the equivalent expression to \( 3^7 \cdot 3^{-9} \) with only positive exponents is:

\[ \frac{1}{9} \]

The final answer is:

\[ \frac{1}{9} \]