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Chapter 7: Problem 8

Convert each number into scientific notation. $$400,000$$

Short Answer

Expert verified
4.0 \times 10^5

Step by step solution

01

Identify the significant figures

Identify the non-zero digits in the number. For 400,000, the significant figures are 4.
02

Place the decimal point

Place the decimal point after the first significant figure to transform the number into a value between 1 and 10. So, 400,000 becomes 4.00000.
03

Count the number of places moved

Count how many places the decimal point was moved to get from the original number to the form in step 2. The decimal point in 400,000 is moved 5 places to the left.
04

Write in scientific notation

Combine the significant figure and the power of 10. Remember, you moved the decimal 5 places to the left, which is expressed as 10 raised to the power of the number of places moved. Thus, 400,000 is written as \[4.0 \times 10^5\].

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Key Concepts

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

Significant Figures
Significant figures are the digits in a number that carry meaningful information about its precision. In the context of scientific notation, identifying significant figures is crucial because we need to know how many meaningful digits we are dealing with. For the number 400,000, the significant figures are 4. This is because the only non-zero digit in 400,000 is 4. However, in other numbers like 40.32, all the digits 4, 0, 3, and 2 are significant. The non-zero digits and any zeros between them or after the decimal point (if they follow significant digits) are considered significant.
Decimal Point Placement
Placing the decimal point correctly is a vital step in converting a number into scientific notation. The goal is to transform the number so that it lies between 1 and 10. For 400,000, we place the decimal point after the first significant figure (which is 4). This gives us 4.00000. It鈥檚 important to note that zeros to the right of the decimal point after a significant digit are not relevant unless stated, hence 4.00000 can be simplified to 4. The process of moving the decimal point helps standardize large or small numbers into a manageable form for calculations.
Powers of 10
The *powers of 10* part of scientific notation lets us express how much the decimal point has been moved. For 400,000, we move the decimal point 5 places to the left. This is represented mathematically as \(10^5\). It indicates that the original number is 4 multiplied by ten to the power of 5. If you were working with a small number, for example, 0.0004, moving the decimal point 4 places to the right would shift the power of 10 to \(10^{-4}\). Understanding and using powers of 10 is essential for simplifying and working with very large or very small numbers efficiently.
Number Conversion
Number conversion into scientific notation is the process of simplifying numbers for ease of use and readability. The conversion has several steps:
  • Identify the significant figures.
  • Place the decimal point to form a value between 1 and 10.
  • Determine the number of places the decimal moved.
  • Combine the significant figure with the power of 10.
So, to convert 400,000, we identified 4 as the significant figure, moved the decimal point 5 places to the left transforming it to 4.0, and represented the number as \(4.0 \times 10^5\).

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Most popular questions from this chapter

Multiply out each of the following. As you work out the problems, identify those exercises that are either a perfect square or the difference of two squares. $$(3 x+y)^{2}$$

Estimate the answer without actually carrying out the computation and make the most appropriate choice. If you multiply 0.00032 by \(420,000,\) the result is closest to (a) 0.12 (b) 1.2 (c) 12 (d) 120 (e) 1200

In this section we defined \(a^{0}=1 .\) It is important to realize that a definition is neither right nor wrong-it just is. The proper question to ask about a definition is "Is it useful?" What happens if someone decides on an "alternative definition" such as \(a^{0}=7\) because 7 happens to be his or her favorite number? What happens when we consider the expression \(a^{0} \cdot a^{4} ?\) If you use exponent rule 1 you get \(a^{0} \cdot a^{4}=a^{0+4}=a^{4} .\) However, if you use this alternative definition, you get \(a^{0} \cdot a^{4} \stackrel{\underline{2}}{=} 7 a^{4} .\) The answer we get from the alternative definition is not consistent with the answer we get from the exponent rule. The exponent rules and this alternative definition cannot coexist. Since we do not want to throw away all the exponent rules, we must modify the alternative definition so that it is consistent with all the exponent rules. Verify that our definition of \(a^{0}=1\) is consistent with all five exponent rules.

Multiply out each of the following. As you work out the problems, identify those exercises that are either a perfect square or the difference of two squares. $$(2 a-5)^{2}$$

The length of a rectangle is 3 more twan twice its width. Using \(W\) as the variable, express the perimeter and area of the rectangle in terms of \(W\)
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