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)
37⋅3−9=

Answer 1

To generate equivalent expressions using the properties of integer exponents, we start with the expression \( 3^7 \cdot 3^{-9} \).

1. Apply the Property of Exponents: According to the property \( a^m \cdot a^n = a^{m+n} \), we can combine the exponents:

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

2. Convert to Positive Exponent: To express \( 3^{-2} \) with a positive exponent, we use the property \( a^{-m} = \frac{1}{a^m} \):

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

3. Calculate \( 3^2 \):

   \[ 3^2 = 9 \]

4. Resulting Expression:

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

The simplified fraction with no remaining exponents is:

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