Question

rewrite the following statement using a number in scientific notation.

the number of different seven-character passwords that can be made with 26 letters and 10 numerals is approximately 78 billion.

the number of different seven-character passwords that can be made with 26 letters and 10 numerals is approximately
(use the multiplication symbol in the math palette as needed)

Explanation:

Identify the target number

We need to rewrite the number "78 billion" using scientific notation.

Convert the word to standard decimal form

In the standard numbering system, one billion is represented as \(1,000,000,000\) or \(10^9\).
Therefore, 78 billion is written in standard decimal form as:

$$78,000,000,000$$

Apply scientific notation rules

A number is in scientific notation when it is written in the form \(a \times 10^n\), where \(1 \le |a| < 10\) and \(n\) is an integer.
To convert \(78,000,000,000\) to this form:

1. Place the decimal point after the first non-zero digit to find \(a\):

$$a = 7.8$$

1. Count the number of places the decimal point must move from its original position at the end of the number to its new position between 7 and 8. Moving the decimal point 10 places to the left gives:

$$n = 10$$

Write the final scientific notation expression

Combining our values for \(a\) and \(n\), we get:

$$7.8 \times 10^{10}$$

Answer:

The number of different seven-character passwords that can be made with 26 letters and 10 numerals is approximately <blank>\(7.8 \times 10^{10}\)</blank>.