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Chapter 10: Problem 33

For Exercises \(33-48,\) convert to scientific notation. $$20,000,000$$

Short Answer

Expert verified
The scientific notation is \(2.0 \times 10^7\).

Step by step solution

01

- Identify the Significant Figures

Identify the non-zero digits in the number. In this case, the significant figures in 20,000,000 are '2.'
02

- Determine the Position of the Decimal Point

Place the decimal point after the first significant figure. Thus, 2 becomes 2.0.
03

- Count the Number of Places the Decimal Moves

Count the number of places you move the decimal point to get from the original number to the converted number. Here, the decimal point moves 7 places to the left.
04

- Express the Number in Scientific Notation

Combine the significant figure and the exponent of 10. Since we moved the decimal 7 places to the left, the exponent is 7. Therefore, 20,000,000 can be written as \(2.0 \times 10^7\).

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Key Concepts

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

Significant Figures
Significant figures are crucial in scientific notation as they reflect the precision of the number. In scientific notation, we only consider the digits that provide meaningful information about the precision of the measurement. For instance, in the number 20,000,000, the only significant figure is '2'. Zeros that come after the significant figure but before the decimal point in this context do not add precision and are not considered significant figures. More precisely:
  • The digit '2' is significant because it carries the actual value.
  • Trailing zeros in whole numbers are not considered unless there is a specified decimal point.
Thus, in 20,000,000, only the 2 is significant. Understanding this can help you in accurately working with scientific notation.
Decimal Point
The decimal point is essential for converting numbers into scientific notation. When you convert a large number into scientific notation, you need to place the decimal point right after the first significant figure. In the example of 20,000,000, placing the decimal point after '2' turns it into 2.0. Consider these points:
  • The decimal point is moved to ensure that the resultant number lies between 1 and 10.
  • Every movement of the decimal point affects the exponent in the notation.
This process makes it easier to handle very large or very small numbers because it normalizes them into a format that is easily understood and used.
Exponent
The exponent in scientific notation represents how many times the base number, usually 10, is multiplied by itself. In our example of 20,000,000, the decimal point moves 7 places to the left to convert it into 2.0. Thus, the exponent is 7. This is because:
  • Exponent shows how many decimal positions the point has shifted.
  • A positive exponent means the original number is large (greater than 1), while a negative exponent means the number is small (less than 1).
So, 20,000,000 becomes \(2.0 \times 10^7\), clearly showing that the decimal was moved 7 places to the left. Mastering this concept helps in quickly and effectively dealing with extensive computations.

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Most popular questions from this chapter

Multiply the polynomials. $$\begin{array}{r}9 a^{2}+2 a-4 \\\\\times \quad 4 a^{2}+a+3 \\\\\hline\end{array}$$

Simplify the expression. $$\left(\frac{7 c}{d^{3}}\right)^{2}$$

Multiply the polynomials. $$\left(5 y^{2}-3 y-4\right)\left(y^{2}+4 y+7\right)$$

Simplify the expression. $$\left(9^{3}\right)^{8}$$

Simplify. Write the answers with positive exponents only. $$\frac{x^{5}}{x^{-7}}$$
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