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Chapter 1: Problem 23

Convert these to nonscientific notation: (a) \(1.52 \times\) \(10^{4},\) (b) \(7.78 \times 10^{-8}\)

Short Answer

Expert verified
(a) \(1.52 \times 10^{4}\) in nonscientific notation is 15200, and (b) \(7.78 \times 10^{-8}\) in nonscientific notation is 0.0000000778.

Step by step solution

01

Conversion of (a)

The number \(1.52 \times 10^{4}\) has a positive exponent, indicating that it is greater than 1. Convert this by moving the decimal point to the right according to the value of the exponent, which is 4 in this case. That results in \(15,200\).
02

Conversion of (b)

The number \(7.78 \times 10^{-8}\) has a negative exponent, indicating that it is a fraction, less than 1. Convert this by moving the decimal point to the left according to the value of the exponent, which is 8 in this case. This results in \(0.0000000778\). Remember to pad with zeros accordingly.

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Key Concepts

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

Positive Exponents
When the exponent in scientific notation is positive, it signifies that the number is greater than 1. This is a key concept to grasp. The positive exponent tells us how many places to move the decimal point to the right.
For instance, in the number \(1.52 \times 10^{4}\), the exponent is 4. This means you move the decimal point 4 places to the right. It鈥檚 like multiplying by 10, four times in a row. So 1.52 becomes 15200.
  • Start with the original number: 1.52
  • Since the exponent is 4, move the decimal 4 places right
  • Fill in any gaps with zeros as needed to get 15,200

Positive exponents are straightforward鈥攖hink of them as multiplying, which makes a smaller number larger.
Negative Exponents
Negative exponents are the opposite of positive ones. They indicate that the number is a fraction, less than 1. With negative exponents, the decimal point moves to the left.
Consider the example \(7.78 \times 10^{-8}\). The exponent \(-8\) means you shift the decimal point left by 8 places. This involves adding leading zeros to help move the decimal.
  • Start with the original number: 7.78
  • Move the decimal 8 places to the left, as the exponent is \(-8\)
  • Insert the necessary zeros to form 0.0000000778

Understanding negative exponents is crucial鈥攊t's like dividing or making the number smaller. Each move left makes the number one-tenth as big as before.
Decimal Point Conversion
Decimal point conversion is the process of shifting the decimal point to represent the number in standard decimal form from scientific notation. This skill is essential for interpreting scientific data in everyday numbers.
To convert a number in scientific notation to decimal form, assess the number of spaces to move the decimal:
  • Positive exponent: Move the decimal to the right
  • Negative exponent: Move it to the left
  • In both cases, ensure to adjust by adding zeros where gaps occur

Utilizing these rules, scientific notation becomes less intimidating and much more manageable, allowing you to translate scientific numbers into forms you deal with every day.

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Most popular questions from this chapter

Magnesium (Mg) is a valuable metal used in alloys, in batteries, and in chemical synthesis. It is obtained mostly from seawater, which contains about \(1.3 \mathrm{~g}\) of Mg for every kilogram of seawater. Calculate the volume of seawater (in liters) needed to extract \(8.0 \times 10^{4}\) tons of \(\mathrm{Mg}\), which is roughly the annual production in the United States. (Density of seawater = \(1.03 \mathrm{~g} / \mathrm{mL} .)\)

Tums is a popular remedy for acid indigestion. A typical Tums tablet contains calcium carbonate plus some inert substances. When ingested, it reacts with the gastric juice (hydrochloric acid) in the stomach to give off carbon dioxide gas. When a 1.328 -g tablet reacted with \(40.00 \mathrm{~mL}\) of hydrochloric acid (density: \(1.140 \mathrm{~g} / \mathrm{mL}\) ), carbon dioxide gas was given off and the resulting solution weighed 46.699 g. Calculate the number of liters of carbon dioxide gas released if its density is \(1.81 \mathrm{~g} / \mathrm{L}\).

Write the equations for converting degrees Celsius to degrees Fahrenheit and degrees Fahrenheit to degrees Celsius.

What is the number of significant figures in each of these measured quantities? (a) \(40.2 \mathrm{~g} / \mathrm{cm}^{3}\), (b) \(0.0000003 \mathrm{~cm}\) (c) \(70 \mathrm{~min}\) (d) \(4.6 \times 10^{19}\) atoms.

This procedure was carried out to determine the volume of a flask. The flask was weighed dry and then filled with water. If the masses of the empty flask and the filled flask were \(56.12 \mathrm{~g}\) and \(87.39 \mathrm{~g},\) respectively, and the density of water is \(0.9976 \mathrm{~g} / \mathrm{cm}^{3},\) calculate the volume of the flask in cubic centimeters.
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