Apply the Properties of Integer Exponents to generate equivalent expressions to 3 superscript7⋅3−superscript9

Question

with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.(1 point)
37⋅3−9=

Answer 1

To simplify the expression \( 3^7 \cdot 3^{-9} \) using the properties of integer exponents, we can apply the property that states \( a^m \cdot a^n = a^{m+n} \).

1. Combine the exponents:

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

2. Convert to positive exponents:

To express \( 3^{-2} \) with a positive exponent, we use the property \( a^{-n} = \frac{1}{a^n} \):

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

3. Calculate \( 3^2 \):

\[ 3^2 = 9 \]

4. Write the final answer:

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

Thus, the simplified expression \( 3^7 \cdot 3^{-9} \) equals \( \frac{1}{9} \).