Let's solve the given expression step by step.
We start with the expression:
[tex]\[9^9 \cdot 9^{-6}\][/tex]
### Step 1: Simplify the expression using properties of exponents
One of the key properties of exponents is that when multiplying like bases, you add the exponents. Therefore:
[tex]\[9^9 \cdot 9^{-6} = 9^{9 - 6} = 9^3\][/tex]
### Step 2: Calculate \(9^3\)
Next, we need to find the value of \(9^3\).
[tex]\[9^3 = 9 \times 9 \times 9\][/tex]
Let's break this down:
[tex]\[9 \times 9 = 81\][/tex]
[tex]\[81 \times 9 = 729\][/tex]
Thus:
[tex]\[9^3 = 729\][/tex]
### Step 3: Verify the components individually
For completeness, let's also look at the components of the original expression.
First, we calculate \(9^9\):
[tex]\[9^9 = 387420489\][/tex]
Next, we look at \(9^{-6}\), which is the reciprocal of \(9^6\):
[tex]\[9^6 = 531441\][/tex]
[tex]\[9^{-6} = \frac{1}{531441} \approx 1.8816764231589208 \times 10^{-6}\][/tex]
### Step 4: Multiply the results from Step 3
Finally, if we multiply the simplified values from Step 1:
[tex]\[387420489 \times 1.8816764231589208 \times 10^{-6} = 729.0\][/tex]
### Conclusion
Combining all the parts, the numerical equivalent of \(9^9 \cdot 9^{-6}\) is:
[tex]\[729.0\][/tex]
So, the final answer is:
[tex]\[ \boxed{729.0} \][/tex]
