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Chapter 10: Problem 63

For Exercises \(49-64\), convert to standard form. $$9.001 \times 10^{8}$$

Short Answer

Expert verified
900,100,000

Step by step solution

01

Understand Scientific Notation

The given number is in scientific notation, which expresses numbers as a product of a coefficient and a power of 10.
02

Identify the Coefficient and Exponent

Here, the coefficient is 9.001 and the exponent is 8, as seen in the expression: \[9.001 \times 10^{8}\].
03

Expand the Scientific Notation

To convert to standard form, move the decimal point of the coefficient 8 places to the right, because the exponent is positive. This gives: \[9.001 \rightarrow 900,100,000\].

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Key Concepts

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

Standard Form Conversion
When converting from scientific notation to standard form, you need to understand the relationship between the coefficient and the exponent. Scientific notation is useful for expressing very large or very small numbers in a concise way. To convert a number in scientific notation to standard form, you follow a few simple steps:
  • Identify the coefficient and the exponent in the number.
  • Move the decimal point in the coefficient to the right or left, depending on whether the exponent is positive or negative.
  • If the exponent is positive, the decimal point is moved to the right.
  • If the exponent is negative, the decimal point is moved to the left.
These steps will help you accurately translate the scientific notation to a standard number format. In our given exercise, the number \(9.001 \times 10^{8}\) is transformed to its standard form as 900,100,000 by moving the decimal point 8 places to the right.
Coefficient and Exponent
The coefficient and the exponent are the core components of a number in scientific notation.
The coefficient is the number that appears before the multiplication sign, while the exponent indicates the power of 10 to which the coefficient is raised. Let鈥檚 break it down:
  • The coefficient must be a number between 1 and 10. For example, in \(9.001 \times 10^{8}\), 9.001 is the coefficient.
  • The exponent shows how many times the coefficient should be multiplied by 10. In the same number, 8 is the exponent.
Proper understanding of these elements will make it clearer how to convert scientific notation to standard form and vice versa. Recognizing these parts helps in performing operations such as addition, subtraction, multiplication, and division with scientific notation as well.
Decimal Point Movement
Decimal point movement is key to converting numbers in scientific notation to standard form. Depending on the exponent, you will move the decimal point of the coefficient to transform the number:
  • If the exponent is positive, move the decimal point to the right.
  • If the exponent is negative, move the decimal point to the left.
For instance, in the exercise given, \(9.001 \times 10^{8}\), the exponent 8 tells us to move the decimal point 8 places to the right, changing 9.001 to 900,100,000.
Remembering these simple rules helps in accurately converting between scientific notation and standard form, making complex calculations easier to handle.

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