Question

Which expression leaves you with the largest number on the number line?

(10 to the second power) to the eighth power
10 to the eighth power divided by 10 to the eighteenth power
10 to the fifteenth power divided by 10 to the negative third power
10 to the fifth power times 10 to the negative fourth times 10 to the negative fifth power

Answer 1

Let's evaluate each expression one by one.

1. **\((10^2)^8\)**:
Using the power of a power rule \((a^m)^n = a^{m \cdot n}\):
\[
(10^2)^8 = 10^{2 \cdot 8} = 10^{16}
\]

2. **\(\frac{10^8}{10^{18}}\)**:
Using the quotient rule \(\frac{a^m}{a^n} = a^{m-n}\):
\[
\frac{10^8}{10^{18}} = 10^{8-18} = 10^{-10}
\]

3. **\(\frac{10^{15}}{10^{-3}}\)**:
Again, using the quotient rule:
\[
\frac{10^{15}}{10^{-3}} = 10^{15 - (-3)} = 10^{15 + 3} = 10^{18}
\]

4. **\(10^5 \times 10^{-4} \times 10^{-5}\)**:
Using the product rule \(a^m \times a^n = a^{m+n}\):
\[
10^5 \times 10^{-4} \times 10^{-5} = 10^{5 + (-4) + (-5)} = 10^{5 - 4 - 5} = 10^{-4}
\]

Now, let's summarize the results:
- Expression 1: \(10^{16}\)
- Expression 2: \(10^{-10}\)
- Expression 3: \(10^{18}\)
- Expression 4: \(10^{-4}\)

Now we can compare the exponents to find which expression leaves us with the largest number:
- \(10^{16}\) (from Expression 1)
- \(10^{-10}\) (from Expression 2)
- \(10^{18}\) (from Expression 3)
- \(10^{-4}\) (from Expression 4)

The largest exponent is clearly **18**, from Expression 3.

Thus, the expression that leaves you with the largest number on the number line is:
\[
\frac{10^{15}}{10^{-3}} = 10^{18}
\]