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Chapter 4: Problem 13
Evaluate: a. |9| b. |-9| c. |-1000| d. \(-|-1000|\)
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Simplify the following expressions by using properties of exponents. Write your final answers with only positive exponents. a. \(\frac{\left(-2 x^{3} y^{-1}\right)^{-3}}{\left(x^{2} y^{-2}\right)^{0}}\) b. \(\frac{\left(-2 x^{3} y^{-1}\right)^{-2}}{\left(x^{2} y^{-2}\right)^{-1}}\) c. \(\left(\frac{3 x^{2} y^{-5}}{5 x^{3} y^{4}}\right)^{-1}\) d. \(\left[\left(3 x^{-1} z^{4}\right)^{-2}\right]^{-3}\)
a. In 2006 Japan had a population of approximately 127.5 million people and a total land area of about 152.5 thousand square miles. What was the population density (the number of people per square mile)? b. In 2006 the United States had a population of approximately 300 million people and a total land area of about 3620 thousand square miles. What was the population density of the United States? c. Compare the population densities of Japan and the United States.
The average distance from Earth to the sun is about \(150,000,000 \mathrm{~km}\), and the average distance from the planet Venus to the sun is about \(108,000,000 \mathrm{~km}\). a. Express these distances in scientific notation. b. Divide the distance from Venus to the sun by the distance from Earth to the sun and express your answer in scientific notation. c. The distance from Earth to the sun is called 1 astronomical unit (1 A.U.) How many astronomical units is Venus from the sun? d. Pluto is \(5,900,000,000 \mathrm{~km}\) from the sun. How many astronomical units is it from the sun?
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.
Without using a calculator show how you can solve for \(x\). a. \(10^{x-5}=1000\) b. \(\log (2 x+10)=2\) c. \(10^{3 x-1}=0.0001\) d. \(\log (500-25 x)=3\)
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