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Chapter 5: Problem 2

Write each of the following in scientific notation. For example \(27800=(2.78)(10)^{4}\). 117

Short Answer

Expert verified
\(1.17 \times 10^2\)

Step by step solution

01

Identify the base number

Look at the given number, 117. We first identify the base number for writing in scientific notation, which must be a number between 1 and 10. For 117, this number is 1.17.
02

Count the decimal place movements

Next, determine how many places the decimal must move to convert 1.17 back to 117. Since 1.17 requires moving the decimal two places to the right to become 117, we will use this count to determine the exponent of 10.
03

Write in scientific notation

Combine the base number with the power of 10. Since we moved the decimal two places, the exponent will be positive 2 (for moving to the left to revert to the original number). Therefore, the scientific notation of 117 is written as \(1.17 \times 10^2\).

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Key Concepts

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

Base Number
In scientific notation, the base number is an essential component. This base number must lie between 1 and 10. Understanding how to choose the correct base number is crucial for writing a number in scientific notation. For example, let's examine the number 117. To identify the base number, we transform 117 into a value between 1 and 10, which results in 1.17. This base number ensures that the scientific notation retains the value of the original number in a standardized format.
  • A base number greater than 10 or less than 1 contradicts the principle of scientific notation.
  • Turning any number into this format involves shifting the decimal point to produce a number within this specified range.
Exponent of 10
Once you have determined the base number, the next step is understanding the role of the exponent in scientific notation. The exponent of 10 reflects how many times the base number must be multiplied by 10 to return to its original value. Using the number 117 again as an example, we transform it to 1.17. The displacement of the decimal point involved two moves to the left, which indicates that the base number must be multiplied by 10 twice, or by \(10^2\), to retrieve the original number.
  • The exponent is positive when moving the decimal to the left and negative when moving it to the right.
  • It shows both the scale and magnitude of the number.
Decimal Place Movement
Understanding the movement of the decimal place is integral to converting numbers into scientific notation. It involves counting how many positions the decimal point moves to change the original number into a base number. For 117, the decimal begins after the 7 and must shift two places left to give 1.17. This movement not only alters the base number but also establishes the exponent.
  • Moving the decimal point determines the size of the exponent, which defines how extensive the scaling factor is.
  • The movement direction (left or right) influences the sign of the exponent, helping to decode the original size of the number.
By successfully noting the decimal movements, you translate any number into an easy-to-compare scientific format. This method simplifies handling very large or very small quantities.

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Most popular questions from this chapter

Write each of the following in scientific notation. For example \(27800=(2.78)(10)^{4}\). 812,000

Use scientific notation and the properties of exponents to help you perform the following operations. \(\sqrt{0.00000009}\)

Use your calculator to estimate each of the following to the nearest one- thousandth. (a) \(7^{\frac{4}{3}}\) (b) \(10^{\frac{4}{5}}\) (c) \(12^{\frac{3}{5}}\) (d) \(19^{\frac{2}{5}}\) (e) \(7^{\frac{3}{4}}\) (f) \(10^{\frac{5}{4}}\)

Perform the indicated operations and express answers in simplest radical form. (See Example 5.) \(\frac{\sqrt[4]{27}}{\sqrt{3}}\)

Use your calculator to evaluate each of the following. Express final answers in ordinary notation. (a) \((27,000)^{2}\) (b) \((450,000)^{2}\) (c) \((14,800)^{2}\) (d) \((1700)^{3}\)(e) \((900)^{4}\) (f) \((60)^{5}\) (g) \((0.0213)^{2}\) (h) \((0.000213)^{2}\) (i) \((0.000198)^{2}\) (j) \((0.000009)^{3}\)
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