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Chapter 2: Problem 106

Convert \(45,800,000\) to scientific notation.

Short Answer

Expert verified
4.58 \times 10^7

Step by step solution

01

Identify the Significant Figures

Determine the significant figures in the number 45,800,000. The significant figures are 4, 5, and 8.
02

Place the Decimal Point

Move the decimal point in the number 45,800,000 to the left until it is just to the right of the first significant figure. This gives 4.58.
03

Count the Number of Places Moved

Count how many places the decimal point was moved to get from 45,800,000 to 4.58. The decimal point was moved 7 places to the left.
04

Write in Scientific Notation

Combine the significant figure placement with the decimal places moved. Since we moved the decimal 7 places to the left, we use a positive exponent: the number in scientific notation is written as \(4.58 \times 10^7\).

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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 converting a number into scientific notation. Significant figures (or digits) are the meaningful numbers in a value that contribute to its precision. For example, in the number \(45,800,000\), the significant figures are 4, 5, and 8 because they are the first non-zero digits. Recognizing these is the first step. These digits help us maintain accuracy when representing the number in scientific notation. When stripping away leading or trailing zeros, we concentrate on the significant figures only.
Decimal Point Movement
Moving the decimal point is an essential process in converting a standard number to scientific notation. For the number \(45,800,000\), we need to place the decimal point after the first significant figure, 4, which gives us \(4.58\). To find out the correct exponent, count the spaces moved from the original decimal point to this new location. In this case, the decimal point moved 7 places to the left. This method helps clarify how large or small the number is, making it easier to handle, especially when dealing with very large or very small values in scientific and mathematical contexts.
Positive Exponent
A positive exponent indicates that the original number is larger than one. After moving the decimal point in the number \(45,800,000\), you count the number of places moved to arrive at the correct exponent. We moved the decimal 7 places to the left, which results in the exponent being \(7\). Therefore, the scientific notation for \(45,800,000\) is written as \(4.58 \times 10^7\). The exponent tells us how many times we need to multiply 4.58 by 10 to get back to the original number. Scientific notation makes it easier to write and read very large numbers by condensing them into a more manageable form.

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