Problem 75 - Beginning and Intermediate Algebra

Question

Write the answer using scientific notation. $$\frac{6.4 \times 10^{-7}}{8.0 \times 10^{6}}$$

Step-by-Step Solution

Answer

\(0.8 \times 10^{-13}\)

Step 1: Divide the coefficients

First, let's divide the coefficients 6.4 and 8.0: \( \frac{6.4}{8} \) The result is: \(0.8\)

Step 2: Divide the powers of 10

Next, we need to divide the powers of 10 from the original expression: \( \frac{10^{-7}}{10^6} \) Since we are dividing the same base (10), we should subtract the exponents: \(10^{-7-6} \) So, the result is: \(10^{-13}\)

Step 3: Combine the results

Now, we will combine the results from Step 1 and Step 2 to get the final answer in scientific notation: \(0.8 \times 10^{-13}\) As the answer is in proper scientific notation, meaning the coefficient is between 1 and 10, this expression is our final answer.

Key Concepts

Dividing Powers of Ten

In scientific notation, we often encounter numbers that are multiplied or divided by powers of ten. To divide powers of ten, it's important to remember that when they share the same base, in this case 10, we can simply subtract the exponents.

Consider the division of two powers of ten:

• For example, \( \frac{10^{-7}}{10^6} \), you subtract the exponents:
• This becomes \(10^{-7 - 6} \).
• The result is \(10^{-13} \).

This rule is derived from properties of exponents and makes dealing with large or small numbers much simpler. By subtracting exponents, we efficiently condense the calculation into a manageable form.

Coefficients in Scientific Notation

When working with scientific notation, the coefficient is a crucial component. It's the number in front of the multiplication sign that

• must be a number between 1 and 10.
• In our example, we had coefficients 6.4 and 8.0.

To divide these coefficients:

• Perform the simple division of \( \frac{6.4}{8} \).
• This results in 0.8.

However, 0.8 is not between 1 and 10. Thus, we need to adjust it to meet the criteria for scientific notation. We can rewrite it as 8.0 and adjust the power of ten accordingly. This ensures the expression adheres to proper scientific notation standards.

Exponents Subtraction

Exponents subtraction is fundamental when working within scientific notation, especially for operations involving division. The core idea is to subtract one exponent from another when dealing with the same base. In our problem, the base is 10.

Here's a concise breakdown:

• Identify the exponents involved, for instance, \(-7\) and \(6\).
• Subtract the second exponent from the first: \(-7 - 6\).
• This gives you \(-13\), a straightforward result leading to \(10^{-13}\).

Subtraction of exponents not only simplifies the division but also maintains the precision of very large or small numbers. This simplification is why scientific notation is such a powerful tool in mathematics.