The problem asks to express the result of the division $\frac{1.5 \times 10^{-37}}{7.5 \times 10^{-111}}$ in scientific notation and to choose the correct answer from the options a) $0.2 \times 10^{-148}$, b) $0.2 \times 10^{74}$, c) $2 \times 10^{73}$, d) $2 \times 10^{-149}$.

1. Problem Description

The problem asks to express the result of the division $\frac{1.5 \times 10^{-37}}{7.5 \times 10^{-111}}$ in scientific notation and to choose the correct answer from the options a) $0.2 \times 10^{-148}$, b) $0.2 \times 10^{74}$, c) $2 \times 10^{73}$, d) $2 \times 10^{-149}$.

2. Solution Steps

First, let's calculate the value of the expression:

$\frac{1.5 \times 10^{-37}}{7.5 \times 10^{-111}} = \frac{1.5}{7.5} \times \frac{10^{-37}}{10^{-111}}$

$\frac{1.5}{7.5} = \frac{15}{75} = \frac{1}{5} = 0.2$

$\frac{10^{-37}}{10^{-111}} = 10^{-37 - (-111)} = 10^{-37 + 111} = 10^{74}$

So, we have

$\frac{1.5 \times 10^{-37}}{7.5 \times 10^{-111}} = 0.2 \times 10^{74}$

Now we need to express this in proper scientific notation, which means the number multiplying the power of 10 should be between 1 and

1

0. $0.2 \times 10^{74} = (2 \times 10^{-1}) \times 10^{74} = 2 \times 10^{-1+74} = 2 \times 10^{73}$

Among the multiple-choice options, the original calculation result $0.2 \times 10^{74}$ is option b).

3. Final Answer

b) $0.2 \times 10^{74}$