Let's solve the expression step by step.
The expression is:
[
\frac{9 \cdot 10^9 \cdot |3 \cdot 10^{-9}| \cdot |6 \cdot 10^{-9}|}{1 \cdot 10^{10^{-2}}}
]
First, we simplify the absolute values. Since both (3 \cdot 10^{-9}) and (6 \cdot 10^{-9}) are positive, we can drop the absolute value signs:
[
= \frac{9 \cdot 10^9 \cdot (3 \cdot 10^{-9}) \cdot (6 \cdot 10^{-9})}{1 \cdot 10^{-2}}
]
Now we can simplify the product in the numerator:
[
= 9 \cdot 3 \cdot 6 \cdot 10^9 \cdot 10^{-9} \cdot 10^{-9}
]
Calculating (9 \cdot 3 \cdot 6):
[
= 9 \cdot 3 = 27
]
[
= 27 \cdot 6 = 162
]
Next, we calculate the powers of (10):
[
10^9 \cdot 10^{-9} \cdot 10^{-9} = 10^{9 - 9 - 9}
]
[
= 10^{-9}
]
Putting it all together in the numerator:
[
= 162 \cdot 10^{-9}
]
Now, we substitute back into the fraction:
[
\frac{162 \cdot 10^{-9}}{1 \cdot 10^{-2}} = \frac{162 \cdot 10^{-9}}{10^{-2}}
]
To divide the powers of (10), we subtract the exponents:
[
= 162 \cdot 10^{-9 - (-2)} = 162 \cdot 10^{-9 + 2} = 162 \cdot 10^{-7}
]
Thus, the simplified result is:
[
162 \cdot 10^{-7}
]
Or expressed in standard scientific notation:
[
= 1.62 \cdot 10^{-5}
]
Therefore, the final answer is:
[
1.62 \cdot 10^{-5}
]
