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Chapter 6: Problem 1

Write each number in scientific notation. a. \(34,000,000,000\) (a) b. \(-2,100,000\) c. 10,060

Short Answer

Expert verified
a. 3.4 脳 10^{10}, b. -2.1 脳 10^{6}, c. 1.006 脳 10^{4}.

Step by step solution

01

- Identify the Coefficient

For each number, we must first identify the coefficient that ranges from 1 to less than 10. This involves moving the decimal point in the number to get a value between these two limits. Each move of the decimal to the left will increase the exponent by one.
02

- Calculating the Exponent for 34,000,000,000

For the number 34,000,000,000, move the decimal 10 places to the left to get 3.4. So, when written in scientific notation, it is 3.4 multiplied by 10 to the power of 10.
03

- Result for 34,000,000,000

The scientific notation of 34,000,000,000 is written as:\[34,000,000,000 = 3.4 \times 10^{10}\]
04

- Calculating the Exponent for -2,100,000

For the number -2,100,000, move the decimal 6 places to the left to get -2.1. Therefore, it is -2.1 multiplied by 10 to the power of 6.
05

- Result for -2,100,000

The scientific notation of -2,100,000 is:\[-2,100,000 = -2.1 \times 10^{6}\]
06

- Calculating the Exponent for 10,060

For 10,060, move the decimal 4 places to the left to get 1.006. So, it is 1.006 multiplied by 10 to the power of 4.
07

- Result for 10,060

The scientific notation for 10,060 is:\[10,060 = 1.006 \times 10^{4}\]

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Key Concepts

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

Understanding Exponents
Exponents are a critical component of scientific notation. They tell us how many times we need to multiply the base number by ten. When you write a number in scientific notation, you're expressing it as a product of the coefficient and a power of ten.
  • The power of ten, or the exponent, indicates how many places the decimal point has shifted to form the coefficient.
  • An exponent of 10, for example, means the decimal has moved 10 places to the left, which makes the number large. Conversely, a negative exponent suggests the decimal moved right, making the number smaller.
In scientific notation, understanding how exponents work is fundamental. They capture both the magnitude and the direction of the decimal shift, facilitating the expression and calculation of both very large and very small numbers.
The Role of the Coefficient
In scientific notation, the coefficient is the number that multiplies by the power of ten. It is always a number greater than or equal to 1 but less than 10.
  • To find the coefficient, we must adjust the original number by moving its decimal point.
  • The goal is to form a new number between 1 and 10. This new number becomes the coefficient.
For example, with the number 34,000,000,000, the coefficient is 3.4 after shifting the decimal 10 places. This process ensures that the scientific notation is standardized across different numbers, simplifying computations and representation.
Adjusting the Decimal Point
The decimal point plays a vital role in converting a number to scientific notation. Shifting the decimal is crucial in determining both the coefficient and the exponent.
  • When moving the decimal point, notice how each position shift corresponds to a change in the exponent.
  • Moving the decimal left increases the exponent, showcasing a larger number, whereas moving it right decreases the exponent, indicating a smaller number.
Take the number -2,100,000 as an example; the decimal moves 6 places to the left to form the coefficient -2.1, paired with an exponent of 6. The careful positioning of the decimal is what allows scientific notation to effectively represent numbers regardless of their size.

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Most popular questions from this chapter

APPLICATION Webster owns a set of antique dining-room furniture that has been in his family for many years. The historical society tells him that furniture similar to his has been appreciating in value at \(10 \%\) per year for the last 20 years and that his furniture could be worth \(\$ 10,000\) now. a. Which letter in the equation \(y=A(1+r)^{x}\) could represent the value of the furniture 20 years ago when it started appreciating? (a) b. Substitute the other given information into the equation \(y=A(1+r)^{x}\). (A) c. Solve your equation in 4 b to find how much Webster's furniture was worth

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