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Chapter 1: Problem 58

Express the following measurements in scientific notation: (a) \(453.32 \mathrm{mg}\) (b) \(0.0000421 \mathrm{~mL}\) (c) \(667,000 \mathrm{~g}\)

Short Answer

Expert verified
a) \(4.5332 \times 10^2\) mg, b) \(4.21 \times 10^{-5}\) mL, c) \(6.67 \times 10^5\) g.

Step by step solution

01

Understanding Scientific Notation

Scientific notation is a way to express very large or very small numbers conveniently. It is composed of a number between 1 and 10 multiplied by a power of ten. For example, the number can be expressed in the form of \( a \times 10^n \), where \( 1 \leq a < 10 \) and \( n \) is an integer.
02

Convert 453.32 mg to Scientific Notation

To express \( 453.32 \) in scientific notation, identify how many places the decimal point should move to make the number between 1 and 10. Move the decimal point 2 places to the left: \( 4.5332 \times 10^2 \) since you move the decimal 2 places, the exponent is 2.
03

Convert 0.0000421 mL to Scientific Notation

To convert \( 0.0000421 \) to scientific notation, move the decimal point to the right until the number is between 1 and 10. Move the decimal point 5 places to get \( 4.21 \times 10^{-5} \). Thus the exponent is -5.
04

Convert 667,000 g to Scientific Notation

For \( 667,000 \), move the decimal point 5 places to the left to express it as a number between 1 and 10: \( 6.67 \times 10^5 \). The exponent is positive because you moved the decimal to the left.

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Key Concepts

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

Measurement Conversion
Measurement conversion is an essential skill for interpreting scientific data. It involves changing a measurement from one unit to another, which allows for easy comparison and calculation in different scientific contexts. When dealing with conversions and scientific notation, it often involves understanding both the numerical value and the magnitude based on units used. For instance:
  • Milligrams to grams requires division by 1000 because there are 1000 milligrams in a gram.
  • Milliliters to liters involves a similar division by 1000, as there are 1000 milliliters in a liter.
  • Grams to kilograms requires dividing by 1000 since there are 1000 grams in a kilogram.

Understanding these conversions help when expressing results in scientific notation, as being aware of the scale of the measurements is crucial for ensuring that calculations correctly reflect their intended values.
Exponent Rules
Exponent rules are core to understanding scientific notation. They are mathematical guides for operating with powers and exponents, simplifying otherwise complex calculations. When expressing a number in scientific notation, you use these rules to precisely scale the number with powers of ten.
  • Multiplying Powers of Ten: When multiplying numbers in scientific notation, add the exponents. For example, \[ (2 \times 10^3) \times (3 \times 10^2) = 6 \times 10^{3+2} = 6 \times 10^5 \].
  • Dividing Powers of Ten: This rule states subtract the exponents when dividing. For instance, \[ \frac{5 \times 10^6}{2 \times 10^3} = 2.5 \times 10^{6-3} = 2.5 \times 10^3 \].
  • Raising Powers to Other Powers: Multiply the exponents when a power is raised to another power, such as \[ (10^2)^3 = 10^{2 \times 3} = 10^6 \].

These rules simplify handling large and small quantities by condensing calculations into simpler exponential expressions.
Decimal Point Movement
Decimal point movement is key when converting numbers into scientific notation. This process involves shifting the decimal point in a number to transform it into a coefficient between 1 and 10. Here's how you do it:
  • Moving the Decimal Right: To express a small number, move the decimal point to the right until reaching a digit between 1 and 10. Each move right results in a negative exponent for ten. For example, \( 0.0000421 \) becomes \( 4.21 \times 10^{-5} \) after moving 5 places.
  • Moving the Decimal Left: For larger numbers, shift the decimal left until it meets the condition of being between 1 and 10. Each left move adds 1 to the positive exponent of ten. For example, in converting \( 667,000 \), the decimal moves 5 spots, resulting in \( 6.67 \times 10^5 \).
  • Preserving Value: The goal is to preserve the original number's value, adjusted to exponential framing. This technique uses base ten as the constant scale.

Mastering decimal point movement will enhance your capability to express any number skillfully in scientific notation.

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Most popular questions from this chapter

Assume that you have two graduated cylinders, one with a capacity of \(5 \mathrm{~mL}\) (a) and the other with a capacity of \(50 \mathrm{~mL}\) (b). Draw a line in each, showing how much liquid you would add if you needed to measure \(2.64 \mathrm{~mL}\) of water. Which cylinder will give the more accurate measurement? Explain.

Vinaigrette salad dressing consists mainly of oil and vinegar. The density of olive oil is \(0.918 \mathrm{~g} / \mathrm{cm}^{3}\), the density of vinegar is \(1.006 \mathrm{~g} / \mathrm{cm}^{3}\), and the two do not mix. If a certain mixture of olive oil and vinegar has a total mass of \(397.8 \mathrm{~g}\) and a total volume of \(422.8 \mathrm{~cm}^{3}\), what is the volume of oil and what is the volume of vinegar in the mixture?

A bag of Hershey's Kisses contains the following information: Serving size: 9 pieces \(=41 \mathrm{~g}\) Calories per serving: 230 Total fat per serving: \(13 \mathrm{~g}\) (a) The bag contains \(2.0\) lbs of Hershey's Kisses. How many Kisses are in the bag? (b) The density of a Hershey's Kiss is \(1.4 \mathrm{~g} / \mathrm{mL}\). What is the volume of a single Hershey's Kiss? (c) How many Calories are in one Hershey's Kiss? (d) Each gram of fat yields 9 Calories when metabolized. What percent of the calories in Hershey's Kisses are derived from fat?

The volume of water used for crop irrigation is measured in acrefeet, where 1 acre-foot is the amount of water needed to cover 1 acre of land to a depth of \(1 \mathrm{ft}\). (a) If there are 640 acres per square mile, how many cubic feet of water are in 1 acre-foot? (b) How many acre-feet are in Lake Erie (total volume \(=116 \mathrm{mi}^{3}\) )?

A vessel contains \(4.67 \mathrm{~L}\) of bromine, whose density is \(3.10 \mathrm{~g} / \mathrm{cm}^{3}\). What is the mass of the bromine in the vessel (in kilograms)?
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