To determine which expression yields the largest number on the number line, let's simplify each option:
A. \((10^{-2})^{-7}\)
Using the power of a power property, we have: \[ (10^{-2})^{-7} = 10^{-2 \cdot -7} = 10^{14} \]
B. \(\frac{10^8}{10^{14}}\)
Using the quotient property, we simplify: \[ \frac{10^8}{10^{14}} = 10^{8 - 14} = 10^{-6} \]
C. \(\frac{10^{18}}{10^8}\)
Using the quotient property: \[ \frac{10^{18}}{10^{8}} = 10^{18 - 8} = 10^{10} \]
D. \(10^5 \cdot 10^{-4} \cdot 10^{11}\)
Using the product property, we combine the exponents: \[ 10^5 \cdot 10^{-4} \cdot 10^{11} = 10^{5 + (-4) + 11} = 10^{5 - 4 + 11} = 10^{12} \]
Now we summarize the results:
- A: \(10^{14}\)
- B: \(10^{-6}\)
- C: \(10^{10}\)
- D: \(10^{12}\)
Comparing these, we have:
\[ 10^{14} > 10^{12} > 10^{10} > 10^{-6} \]
The largest expression is A, which is \(10^{14}\).
Thus, the expression that leaves you with the largest number on the number line is:
\(\boxed{A}\)