To solve the expression

$\frac{288.8 \times 10^{-7.4} + 18.8 \times 101^{-7.59}}{288.8 + 18.8},$

follow these steps:

Step 1: Calculate each term in the numerator

1. Calculate $288.8 \times 10^{-7.4}$: $288.8 \times 10^{-7.4} \approx 288.8 \times 3.981 \times 10^{-8} \approx 1.1506 \times 10^{-5}$

2. Calculate $18.8 \times 101^{-7.59}$: $101^{-7.59} \approx 1.512 \times 10^{-15}$ $18.8 \times 1.512 \times 10^{-15} \approx 2.8425 \times 10^{-14}$

Step 2: Add the terms in the numerator

$1.1506 \times 10^{-5} + 2.8425 \times 10^{-14} \approx 1.1506 \times 10^{-5}$

(The second term is very small compared to the first one and has a negligible effect on the sum.)

Step 3: Calculate the denominator

$288.8 + 18.8 = 307.6$

Step 4: Divide the sum from Step 2 by the denominator

$\frac{1.1506 \times 10^{-5}}{307.6} \approx 3.74 \times 10^{-8}$

So, the result of the expression is approximately $3.74 \times 10^{-8}$.

Would you like further details or have any other questions?

Here are 5 related questions:

1. How do you handle significant figures in scientific notation?
2. What are the rules for adding and multiplying numbers in scientific notation?
3. How does scientific notation simplify calculations with very large or very small numbers?
4. What is the process for converting between standard form and scientific notation?
5. How do logarithms help in simplifying expressions involving exponents?

Tip: When dealing with very small or large numbers, using scientific notation helps maintain accuracy and avoid errors in calculations.