Problem 59 - Elementary Algebra

Question

\(\frac{9 \times 10^{8} \mathrm{~cm}^{2}}{2 \times 10^{5} \mathrm{~cm}}\)

Step-by-Step Solution

Answer

4.5 × 10^3 cm

Step 1 - Simplify the Problem

Start by rewriting the problem: \[\frac{9 \times 10^{8} \text{ cm}^2}{2 \times 10^{5} \text{ cm}}\]

Step 2 - Simplify the Units

Notice that the units of \(\text{cm}^2\) in the numerator and \(\text{cm}\) in the denominator can be simplified: \[\frac{9 \times 10^{8}}{2 \times 10^{5}} \text{ cm}\]

Step 3 - Divide the Coefficients

Divide the coefficients \(9\) by \(2\): \[\frac{9}{2} = 4.5\]

Step 4 - Subtract the Exponents

Subtract the exponent in the denominator from the exponent in the numerator: \[10^8 - 10^5 = 10^3\]

Step 5 - Combine Results

Combine the results from steps 3 and 4: \[4.5 \times 10^3 \text{ cm}\]

Key Concepts

Scientific Notation

Scientific notation is a way to express very large or very small numbers more conveniently. You can convert any number into scientific notation by writing it as the product of a number between 1 and 10 and a power of 10. For example, instead of writing 900,000,000, you can express it as 9 × 10^8. This format makes multiplication and division simpler, especially when dealing with exponents.

Division of Exponents

When you divide numbers in scientific notation, you can simplify the process by dividing the coefficients and then subtracting the exponents. In general, \(\frac{A \times 10^m}{B \times 10^n} = \frac{A}{B} \times 10^{m-n}\).
Let's apply this to the example. We initially have: \[ \frac{9 \times 10^8 \text{ cm}^2}{2 \times 10^5 \text{ cm}} \]
First, divide the coefficients : \[ \frac{9}{2} = 4.5\]
Next, subtract the exponents: \[ 10^8 - 10^5 = 10^3\]
Combine these results, and you get:
\