Question

Question
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.(1 point)
37⋅3−9=

Answer 1

Apply the Properties of Integer Exponents to generate equivalent expressions to 2^2/2^−4
with only positive exponents, then solve the expression. Provide the solution as a whole number.(1 point)
222−4=

Answer 2

To solve the expression \( 3^7 \cdot 3^{-9} \) using the properties of integer exponents, we will first combine the exponents.

The property of exponents that applies here is:
\[
a^m \cdot a^n = a^{m+n}
\]

So, we can combine the exponents:
\[
3^7 \cdot 3^{-9} = 3^{7 + (-9)} = 3^{-2}
\]

Next, we convert \( 3^{-2} \) to a positive exponent. Using the property:
\[
a^{-n} = \frac{1}{a^n}
\]

We can rewrite \( 3^{-2} \) as:
\[
3^{-2} = \frac{1}{3^2}
\]

Now we calculate \( 3^2 \):
\[
3^2 = 9
\]

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

So the equivalent expression with only positive exponents is \( \frac{1}{9} \).

In conclusion, the final answer is:
\[
\frac{1}{9}
\]

Answer 3

To solve the expression \( \frac{2^2}{2^{-4}} \) using the properties of integer exponents, we will first apply the property that allows us to combine exponents in division:

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

Using this property, we can rewrite the expression:

\[
\frac{2^2}{2^{-4}} = 2^{2 - (-4)} = 2^{2 + 4} = 2^6
\]

Next, we calculate \( 2^6 \):

\[
2^6 = 64
\]

Therefore, \( \frac{2^2}{2^{-4}} \) simplifies to:

\[
64
\]

The final answer as a whole number is:

\[
64
\]