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:

$$\frac{1}{9}$$