Question
Apply the Properties of Integer Exponents to generate equivalent expressions to 3 raised to the 7 power⋅3 raised to the −9 power with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.
Answers
GPT-4o mini
To simplify the expression \( 3^7 \cdot 3^{-9} \) using the properties of integer exponents, we can apply the product of powers property, which states that when multiplying two expressions with the same base, we add their exponents:
\[
3^a \cdot 3^b = 3^{a+b}
\]
Applying this to our expression:
\[
3^7 \cdot 3^{-9} = 3^{7 + (-9)} = 3^{7 - 9} = 3^{-2}
\]
Next, we can rewrite \( 3^{-2} \) using positive exponents. The property for negative exponents states that:
\[
x^{-n} = \frac{1}{x^n}
\]
Therefore,
\[
3^{-2} = \frac{1}{3^2}
\]
Now, we calculate \( 3^2 \):
\[
3^2 = 9
\]
Thus, we can write \( 3^{-2} \) as:
\[
3^{-2} = \frac{1}{9}
\]
In conclusion, the expression \( 3^7 \cdot 3^{-9} \) simplifies to:
\[
\frac{1}{9}
\]
So, the final answer is:
\[
\frac{1}{9}
\]
\[
3^a \cdot 3^b = 3^{a+b}
\]
Applying this to our expression:
\[
3^7 \cdot 3^{-9} = 3^{7 + (-9)} = 3^{7 - 9} = 3^{-2}
\]
Next, we can rewrite \( 3^{-2} \) using positive exponents. The property for negative exponents states that:
\[
x^{-n} = \frac{1}{x^n}
\]
Therefore,
\[
3^{-2} = \frac{1}{3^2}
\]
Now, we calculate \( 3^2 \):
\[
3^2 = 9
\]
Thus, we can write \( 3^{-2} \) as:
\[
3^{-2} = \frac{1}{9}
\]
In conclusion, the expression \( 3^7 \cdot 3^{-9} \) simplifies to:
\[
\frac{1}{9}
\]
So, the final answer is:
\[
\frac{1}{9}
\]