91Ó°ÊÓ

An electron weighs about \(10^{-27}\) gram, and a raindrop weighs about \(10^{-3}\) gram. How many times heavier is a raindrop than an electron? How many times lighter is an electron than a raindrop? What is the order-of- magnitude difference?

Change each number to scientific notation, then simplify using rules of exponents. Show your work, recording your final answer in scientific notation. a. \(10 \%\) of 0.00001 b. \(\frac{0.00005}{50,000}\) c. \(\frac{3}{0.006}\) d. \(\frac{8000}{0.0008}\) e. \(\frac{0.0064}{8000}\) f. \(5,000,000 \cdot 40,000\)

Write each of the following in scientific notation: a. 0.00029 d. 0.00000000001 $$ \text { g. }-0.0049 $$ b. 654.456 e. 0.00000245 c. 720,000 $$ \text { f. }-1,980,000 $$

For each equation determine the value of \(x\) that makes it true. a. \(10^{x}=0.000001\) b. \(10^{x}=\frac{1}{1,000,000}\) c. \(\frac{1}{10^{x}}=0.0001\) d. \(10^{-x}=100,000\)

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?

Determine the order-of-magnitude difference in the sizes of the radii for: a. The solar system \(\left(10^{12}\right.\) meters) compared with Earth \(\left(10^{7}\right.\) meters) b. Protons \(\left(10^{-15}\right.\) meter) compared with the Milky Way \(\left(10^{21}\right.\) meters) c. Atoms \(\left(10^{-10}\right.\) meter) compared with neutrons \(\left(10^{-15}\right.\) meter)

The distance from Earth to the sun is approximately 150 million kilometers. If the speed of light is \(3.00 \cdot 10^{5} \mathrm{~km} / \mathrm{sec},\) how long does it take light from the sun to reach Earth? If a solar flare occurs right now, how long would it take for us to see it?

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. \(\left[\left(3 x^{-1} z^{4}\right)^{-2}\right]^{-3}\)

The pH scale measures the hydrogen ion concentration in a liquid, which determines whether the substance is acidic or alkaline. A strong acid solution has a hydrogen ion concentration of \(10^{-1}\) M. One \(\mathrm{M}\) equals \(6.02 \cdot 10^{23}\) particles, such as atoms, ions, molecules, etc., per liter, or 1 mole per liter. \(^{6}\) A strong alkali solution has a hydrogen ion concentration of \(10^{-14} \mathrm{M}\). Pure water, with a concentration of \(10^{-7} \mathrm{M},\) is neutral. The \(\mathrm{pH}\) value is the power without the minus sign, so pure water has a \(\mathrm{pH}\) of \(7,\) acidic substances have a pH less than \(7,\) and alkaline substances have a \(\mathrm{pH}\) greater than 7 . a. Tap water has a pH of 5.8 . Before the industrial age, rain water commonly had a pH of about \(5 .\) With the spread of modern industry, rain in the northeastern United States and parts of Europe now has a \(\mathrm{pH}\) of about \(4,\) and in extreme cases the \(\mathrm{pH}\) is about \(2 .\) Lemon juice has a \(\mathrm{pH}\) of 2.1. If acid rain with a pH of 3 is discovered in an area, how much more acidic is it than preindustrial rain? b. Blood has a pH of 7.4 ; wine has a pH of about 3.4. By how many orders of magnitude is wine more acidic than blood?

Evaluate: a. |9| b. |-9| c. |-1000| d. \(-|-1000|\)