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Chapter 4: Problem 9

Write each of the following in standard decimal form: a. \(7.23 \cdot 10^{5}\) d. \(1.5 \cdot 10^{6}\) b. \(5.26 \cdot 10^{-4}\) e. \(1.88 \cdot 10^{-4}\) c. \(1.0 \cdot 10^{-3}\) f. \(6.78 \cdot 10^{7}\)

Short Answer

Expert verified
a. 723000, d. 1500000, b. 0.000526, e. 0.000188, c. 0.001, f. 67800000

Step by step solution

01

- Understand Scientific Notation

Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form. It is written in the form of \(a \cdot 10^{n}\), where 1 鈮� |a| < 10 and n is an integer.
02

- Converting Positive Exponents

For positive exponents, move the decimal point to the right. For example, \(7.23 \cdot 10^{5}\) means moving the decimal point 5 places to the right.
03

Step 2a: Convert \(7.23 \cdot 10^{5} \)

Move the decimal point 5 places to the right: \[7.23 \rightarrow 723000\]. The result is 723000.
04

Step 2b: Convert \(1.5 \cdot 10^{6} \)

Move the decimal point 6 places to the right: \[1.5 \rightarrow 1500000\]. The result is 1500000.
05

Step 2c: Convert \(6.78 \cdot 10^{7} \)

Move the decimal point 7 places to the right: \[6.78 \rightarrow 67800000\]. The result is 67800000.
06

- Converting Negative Exponents

For negative exponents, move the decimal point to the left. For example, \(5.26 \cdot 10^{-4}\) means moving the decimal point 4 places to the left.
07

Step 3a: Convert \(5.26 \cdot 10^{-4} \)

Move the decimal point 4 places to the left: \[5.26 \rightarrow 0.000526\]. The result is 0.000526.
08

Step 3b: Convert \(1.88 \cdot 10^{-4} \)

Move the decimal point 4 places to the left: \[1.88 \rightarrow 0.000188\]. The result is 0.000188.
09

Step 3c: Convert \(1.0 \cdot 10^{-3} \)

Move the decimal point 3 places to the left: \[1.0 \rightarrow 0.001\]. The result is 0.001.

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Key Concepts

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

Decimal Form
Decimal form is the standard way of writing numbers using the base ten numeral system. It represents both whole numbers and fractional parts using a decimal point. In scientific notation, large and small numbers are represented as a product of a number between 1 and 10, and a power of 10. Converting these scientific notations back to decimal form involves moving the decimal point left or right based on the exponent. Understanding decimal form is crucial for reading and interpreting scientific notation correctly.
Positive Exponents
In scientific notation, positive exponents indicate how many times the base number (between 1 and 10) should be multiplied by 10. This effectively moves the decimal point to the right. For instance, when we have the number:
  • \(7.23 \times 10^5\)
We move the decimal point 5 places to the right, resulting in:
  • 723000
The key is to make sure you move the decimal point the correct number of places to the right and fill in any gaps with zeros.
Negative Exponents
Negative exponents in scientific notation mean the base number should be divided by 10, rather than multiplied, which moves the decimal point to the left. For example:
  • \(5.26 \times 10^{-4}\)
Here, we move the decimal point 4 places to the left, resulting in:
  • 0.000526
Understanding negative exponents properly ensures you can accurately convert small numbers from scientific notation to decimal form by placing the decimal point in the correct position.
Converting Scientific Notation
Converting between scientific notation and decimal form involves knowing whether the exponent is positive or negative. Here鈥檚 a quick guide:
  • Identify the exponent
  • For positive exponents, move the decimal to the right
  • For negative exponents, move the decimal to the left
  • Add zeros to fill in any gaps created by moving the decimal.
Let鈥檚 convert some examples:
  • \(1.5 \times 10^6 \rightarrow 1500000\)
  • \(1.88 \times 10^{-4} \rightarrow 0.000188\)
Practice these steps, and you will get comfortable moving between scientific and decimal notations quickly!

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Most popular questions from this chapter

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(Requires the use of a calculator that can evaluate powers.) If a rope is wound around a wooden pole, the number of pounds of frictional force, \(F,\) between the pole and the rope is a function of the number of turns, \(N\), according to the equation \(F=14 \cdot 10^{0.70 \mathrm{~N}}\). What is the frictional force when the number of turns is: a. 0.5 b. 1 c. 3

The concentration of hydrogen ions in a water solution typically ranges from \(10 \mathrm{M}\) to \(10^{-15} \mathrm{M}\). (One \(\mathrm{M}\) equals \(6.02 \cdot 10^{23}\) particles, such as atoms, ions, molecules, etc., per liter or 1 mole per liter.) Because of this wide range, chemists use a logarithmic scale, called the pH scale, to measure the concentration (see Exercise 12 of Section 4.6 ). The formal definition of \(\mathrm{pH}\) is \(\mathrm{pH}=-\log \left[\mathrm{H}^{+}\right],\) where \(\left[\mathrm{H}^{+}\right]\) denotes the concentration of hydrogen ions. Chemists use the symbol \(\mathrm{H}^{+}\) for hydrogen ions, and the brackets [ ] mean "the concentration of." a. Pure water at \(25^{\circ} \mathrm{C}\) has a hydrogen ion concentration of \(10^{-7} \mathrm{M}\). What is the \(\mathrm{pH} ?\) b. In orange juice, \(\left[\mathrm{H}^{+}\right] \approx 1.4 \cdot 10^{-3} \mathrm{M}\). What is the \(\mathrm{pH}\) ? c. Household ammonia has a pH of about \(11.5 .\) What is its \(\left[\mathrm{H}^{+}\right] ?\) d. Does a higher pH indicate a lower or a higher concentration of hydrogen ions? e. A solution with a \(\mathrm{pH}>7\) is called basic, one with a \(\mathrm{pH}=7\) is called neutral, and one with a \(\mathrm{pH}<7\) is called acidic. Identify pure water, orange juice, and household ammonia as either acidic, neutral, or basic. Then plot their positions on the accompanying scale, which shows both the \(\mathrm{pH}\) and the hydrogen ion concentration.

Rewrite the following equations using logs instead of exponents. Estimate a solution for \(x\) and then check your estimate with a calculator. Round the value of \(x\) to three decimal places. a. \(10^{x}=12,500\) b. \(10^{x}=3,526,000\) c. \(10^{x}=597\) d. \(10^{x}=756,821\)

In 2006 the United Kingdom generated approximately 81 terawatt-hours of nuclear energy for a population of about 60.6 million on 94,525 miles \(^{2}\). In the same year the United States generated approximately 780 terawatt-hours of nuclear energy for a population of about 300 million on 3,675,031 miles \(^{2}\). A terawatt is \(10^{12}\) watts. a. How many terawatt-hours is the United Kingdom generating per person? How many terawatt-hours is it generating per square mile? Express each in scientific notation. b. How many terawatt-hours is the United States generating per person? How many terawatt-hours are we generating per square mile? Express each in scientific notation. c. How much nuclear energy is being generated in the United Kingdom per square mile relative to the United States? d. Write a brief statement comparing the relative magnitude of generation of nuclear power per person in the United Kingdom and the United States.
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