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Chapter 6: Problem 550

Write each number in scientific notation. $$57,000$$

Short Answer

Expert verified
57,000 = 5.7 \times 10^4

Step by step solution

01

- Identify the significant figures

Identify the significant figures in the number 57,000. The significant figures are the non-zero digits that form the number, which in this case are 5 and 7.
02

- Place the decimal point

Place the decimal point after the first significant figure to form a number between 1 and 10. For 57,000, place the decimal point after the 5 to get 5.7.
03

- Count the number of decimal places

Count how many places the decimal point has moved from its original position. For 57,000, moving the decimal point from 57,000 to 5.7 involves 4 places to the left.
04

- Write in scientific notation

Write the number in scientific notation by combining the significant figure (5.7) with a power of 10 indicating the number of places the decimal point moved. Since the decimal point moved 4 places left, the power of 10 is 4. Thus, 57,000 in scientific notation is written as: \[57,000 = 5.7 \times 10^4\]

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Key Concepts

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

headline of the respective core concept
Understanding significant figures is crucial when working with scientific notation. Significant figures are the digits that carry meaning in a number, contributing to its precision. For the number 57,000, the significant figures are 5 and 7 because they are the non-zero digits. In scientific notation, we use these key digits to represent the number in a more compact form. Always remember, zeros between non-zero digits, or trailing zeros in a decimal number, also count as significant.
headline of the respective core concept
The decimal point is essential in scientific notation to create a number between 1 and 10. Placing the decimal point correctly is necessary for converting a number to scientific notation. For the number 57,000, placing the decimal point after the first significant figure (5) results in the number 5.7. This step ensures accuracy and allows for the simplification of larger or smaller numbers into a standard format.
headline of the respective core concept
Powers of 10 are used in scientific notation to indicate how many places the decimal point has moved. When converting 57,000 to scientific notation, we move the decimal point 4 places left to change it from 57,000 to 5.7. This movement is represented by multiplying by a power of 10. In this case, since we moved the decimal 4 places, we use \(10^4\). Thus, 57,000 is written as \[57,000 = 5.7 \times 10^4\]]. Understanding this concept allows for the efficient and clear representation of both very large and small numbers.

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