Let's tackle both questions one at a time.
Question 1: Exponent Expression
We need to simplify the expression:
\[ (9^{-2} \cdot 9^{7})^{-3} \]
Using the properties of exponents, we can first simplify inside the parentheses:
\[ 9^{-2} \cdot 9^{7} = 9^{-2 + 7} = 9^{5} \]
Now we substitute this back into the expression:
\[ (9^{5})^{-3} = 9^{5 \cdot (-3)} = 9^{-15} \]
Now, to express \(9^{-15}\) in the form requested, we can convert it to a positive exponent:
\[ 9^{-15} = \frac{1}{9^{15}} \]
So, the equivalent expression is:
\[ \frac{1}{9^{15}} \]
Question 2: Distance Between Points
We need to find the distance between the points \((-1, -4)\) and \((-9, -8)\). We can use the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting in our points:
- \(x_1 = -1\), \(y_1 = -4\)
- \(x_2 = -9\), \(y_2 = -8\)
Calculating the differences:
\[ x_2 - x_1 = -9 - (-1) = -9 + 1 = -8 \implies (-8)^2 = 64 \]
\[ y_2 - y_1 = -8 - (-4) = -8 + 4 = -4 \implies (-4)^2 = 16 \]
Now, substituting these into the distance formula:
\[ d = \sqrt{64 + 16} = \sqrt{80} \]
Final Answers
- The expression that is equivalent to the quantity is:
\[ \frac{1}{9^{15}} \]
- The distance between the points \((-1, -4)\) and \((-9, -8)\) is:
\[ \sqrt{80} \text{ units} \]