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Chapter 6: Problem 74

Convert to scientific notation: $$ 391,000,000 $$

Short Answer

Expert verified
3.91 \times 10^8

Step by step solution

01

Identify the Significant Figures

First, identify the significant figures in the number. In 391,000,000, the significant figures are 3, 9, and 1.
02

Place the Decimal Point

Place the decimal point after the first significant figure to get the number in a format where it is between 1 and 10. This gives us 3.91.
03

Count the Number of Places Moved

Count how many places the decimal point has moved from its original position in 391,000,000 to its new position (3.91). The decimal point originally is after the last zero, so it has moved 8 places to the left.
04

Write the Number in Scientific Notation

Write the number in the form of \[ N \times 10^n \] where N is the number obtained in Step 2 and n is the number obtained in Step 3. Therefore, 391,000,000 in scientific notation is \[ 3.91 \times 10^8 \]

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Key Concepts

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

Significant Figures
Understanding significant figures is crucial when working with scientific notation. Significant figures are the digits in a number that are meaningful in terms of accuracy and precision.
In our example, 391,000,000, the significant figures are 3, 9, and 1. These figures are important because they represent the precise parts of the number.
Zeros can be tricky: leading zeros are not significant, but zeros between digits and trailing zeros in decimal points usually are.
By identifying the significant figures, we ensure the accuracy of our scientific notation transformation.
Decimal Point
Placing the decimal point correctly is a key step in converting large numbers to scientific notation.
In scientific notation, we place the decimal point after the first significant figure. This helps us create a number between 1 and 10, which is easier to work with.
For 391,000,000, we place the decimal after 3 to get 3.91.
This step simplifies handling the number, making complex calculations more manageable.
Exponents
Exponents in scientific notation indicate how many places the decimal point has moved.
It's written as a power of 10. To find the exponent, count the positions the decimal point moved.
For example, in 391,000,000, the decimal point moves 8 places to the left to get from 391,000,000 to 3.91.
Therefore, the exponent is 8, and the scientific notation becomes 3.91 脳 10^8.
Large Numbers
Scientific notation is especially useful for representing large numbers efficiently.
Writing 391,000,000 is cumbersome and prone to errors. Using 3.91 脳 10^8 instead simplifies it.
This notation makes it easier to comprehend large quantities, perform arithmetic operations, and ensure precision.
It also helps in comparing large numbers quickly by looking at the exponents.

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