To determine which expression yields the largest number, let's evaluate each of them step by step.
-
\( 105 \cdot 10^{-4} \cdot 10^{11} \):
- First, combine the powers of 10: \[ 10^{-4} \cdot 10^{11} = 10^{-4 + 11} = 10^{7}. \]
- Now, multiply by 105: \[ 105 \cdot 10^{7} = 1.05 \cdot 10^{9} = 1.05 \times 10^{9}. \]
-
\( \frac{10^{18}}{10^{8}} \):
- When dividing powers of 10, subtract the exponents: \[ 10^{18 - 8} = 10^{10}. \]
-
\( \frac{10^{8}}{10^{14}} \):
- Again, subtract the exponents: \[ 10^{8 - 14} = 10^{-6}. \]
- This results in a very small number, specifically \(0.000001\).
-
\( (10^{-2})(-7) \):
- First, evaluate \(10^{-2}\): \[ 10^{-2} = 0.01. \]
- Now, multiply: \[ 0.01 \cdot (-7) = -0.07. \]
- This results in a negative value.
Now, let's summarize the results:
- \( 105 \cdot 10^{-4} \cdot 10^{11} \) = \( 1.05 \times 10^{9} \)
- \( \frac{10^{18}}{10^{8}} \) = \( 10^{10} \) = \( 10,000,000,000 \)
- \( \frac{10^{8}}{10^{14}} \) = \( 10^{-6} \) = \( 0.000001 \)
- \( (10^{-2})(-7) \) = \( -0.07 \)
Comparing all the numbers obtained, the largest number is:
\( 10^{10} \) or \( \frac{10^{18}}{10^{8}} \).