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Chapter 5: Problem 61

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

Short Answer

Expert verified
4.5 \times 10^{-3} \text{ cm}

Step by step solution

01

- Understand the Problem

The goal is to simplify the expression \(\frac{9 \times 10^{-8} \text{ cm}^2}{2 \times 10^{-5} \text{ cm}}\). This involves dividing two numbers that are in scientific notation.
02

- Divide the Coefficients

Divide the coefficients of the numbers separately: \(\frac{9}{2} = 4.5\).
03

- Subtract the Exponents

Subtract the exponent of the denominator from the exponent of the numerator in scientific notation: \(-8 - (-5) = -8 + 5 = -3\).
04

- Combine the Results

Combine the results from the previous steps: the coefficient \(4.5\) and the exponent \(-3\). The final answer is \(4.5 \times 10^{-3} \text{ cm}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Scientific Notation
Scientific notation is a method to express very large or very small numbers in a more compact form. It uses powers of ten to simplify the representation.

For example, instead of writing 0.00000009, we write it as \(9 \times 10^{-8}\).

In scientific notation, a number is written as the product of a coefficient and a power of ten. The coefficient must be a number between 1 and 10, and the exponent on 10 shows how many places to move the decimal point.

This notation is very useful in sciences and engineering, where you frequently encounter extremely large or small quantities.
Exponent Rules
Exponent rules help us manipulate powers of ten in a straightforward manner.

When working with numbers in scientific notation, you will often use these rules:
  • Multiplying powers of ten: \(10^a \times 10^b = 10^{a+b}\)
  • Dividing powers of ten: \(10^a \times 10^b = 10^{a-b}\)
  • Power of a power: \((10^a)^b = 10^{a \times b}\)

Knowing these rules makes it much easier to handle numbers in scientific notation. They simplify calculations that might otherwise be very complex.
Divide Coefficients
To divide numbers in scientific notation, start by dividing their coefficients.

For example, to divide \( \frac{9 \times 10^{-8}}{2 \times 10^{-5}}\), first focus on \(\frac{9}{2}\).

Doing this division yields \(4.5\).

Separating this step simplifies the overall process and prepares you for the next step involving exponents.
Subtract Exponents
When dividing numbers in scientific notation, after dividing the coefficients, the next step is to subtract the exponents.

Take the exponents from the numerator and the denominator and subtract them: \(-8 - (-5)\) which simplifies to \(-8 + 5 = -3\).

Subtraction of exponents is straightforward, and being careful with positive and negative signs ensures accuracy.

Combining this result with the divided coefficients gives you the final answer in scientific notation. For the given example, you get \(4.5 \times 10^{-3}\).

This methodical approach simplifies the process and makes handling scientific notation manageable.

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