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Rules of Exponents
For nonzero real numbers $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ and integers $\text{\hspace{0.17em}}m\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ | |
Product rule | ${a}^{m}\cdot {a}^{n}={a}^{m+n}$ |
Quotient rule | $\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}$ |
Power rule | ${\left({a}^{m}\right)}^{n}={a}^{m\cdot n}$ |
Zero exponent rule | ${a}^{0}=1$ |
Negative rule | ${a}^{-n}=\frac{1}{{a}^{n}}$ |
Power of a product rule | ${\left(a\cdot b\right)}^{n}={a}^{n}\cdot {b}^{n}$ |
Power of a quotient rule | ${\left(\frac{a}{b}\right)}^{n}=\frac{{a}^{n}}{{b}^{n}}$ |
Is $\text{\hspace{0.17em}}{2}^{3}\text{\hspace{0.17em}}$ the same as $\text{\hspace{0.17em}}{3}^{2}?\text{\hspace{0.17em}}$ Explain.
No, the two expressions are not the same. An exponent tells how many times you multiply the base. So $\text{\hspace{0.17em}}{2}^{3}\text{\hspace{0.17em}}$ is the same as $\text{\hspace{0.17em}}2\times 2\times 2,$ which is 8. $\text{\hspace{0.17em}}{3}^{2}\text{\hspace{0.17em}}$ is the same as $\text{\hspace{0.17em}}3\times 3,$ which is 9.
When can you add two exponents?
What is the purpose of scientific notation?
It is a method of writing very small and very large numbers.
Explain what a negative exponent does.
For the following exercises, simplify the given expression. Write answers with positive exponents.
${15}^{\mathrm{-2}}$
${3}^{2}\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}{3}^{3}$
243
${4}^{4}\xf74$
${(5-8)}^{0}$
${6}^{5}\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}{6}^{\mathrm{-7}}$
${5}^{\mathrm{-2}}\xf7{5}^{2}$
For the following exercises, write each expression with a single base. Do not simplify further. Write answers with positive exponents.
${4}^{2}\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}{4}^{3}\xf7{4}^{\mathrm{-4}}$
${4}^{9}$
$\frac{{6}^{12}}{{6}^{9}}$
${\left({12}^{3}\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}12\right)}^{10}$
${12}^{40}$
${10}^{6}\xf7{\left({10}^{10}\right)}^{\mathrm{-2}}$
${7}^{\mathrm{-6}}\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}{7}^{\mathrm{-3}}$
$\frac{1}{{7}^{9}}$
${\left({3}^{3}\xf7{3}^{4}\right)}^{5}$
For the following exercises, express the decimal in scientific notation.
148,000,000
For the following exercises, convert each number in scientific notation to standard notation.
$1.6\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}{10}^{10}$
16,000,000,000
$9.8\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}{10}^{\mathrm{-9}}$
For the following exercises, simplify the given expression. Write answers with positive exponents.
$\frac{m{n}^{2}}{{m}^{\mathrm{-2}}}$
${\left(\frac{{x}^{\mathrm{-3}}}{{y}^{2}}\right)}^{\mathrm{-5}}$
${\left({w}^{0}{x}^{5}\right)}^{\mathrm{-1}}$
${y}^{\mathrm{-4}}{\left({y}^{2}\right)}^{2}$
$\frac{{p}^{\mathrm{-4}}{q}^{2}}{{p}^{2}{q}^{\mathrm{-3}}}$
$\frac{{q}^{5}}{{p}^{6}}$
${(l\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}w)}^{2}$
${\left({y}^{7}\right)}^{3}\xf7{x}^{14}$
$\frac{{y}^{21}}{{x}^{14}}$
${\left(\frac{a}{{2}^{3}}\right)}^{2}$
$\frac{{\left(16\sqrt{x}\right)}^{2}}{{y}^{\mathrm{-1}}}$
$\frac{{2}^{3}}{{\left(3a\right)}^{\mathrm{-2}}}$
$72{a}^{2}$
${\left(m{a}^{6}\right)}^{2}\frac{1}{{m}^{3}{a}^{2}}$
${\left({b}^{\mathrm{-3}}c\right)}^{3}$
$\frac{{c}^{3}}{{b}^{9}}$
${\left({x}^{2}{y}^{13}\xf7{y}^{0}\right)}^{2}$
${\left(9{z}^{3}\right)}^{\mathrm{-2}}y$
$\frac{y}{81{z}^{6}}$
To reach escape velocity, a rocket must travel at the rate of $\text{\hspace{0.17em}}2.2\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}{10}^{6}\text{\hspace{0.17em}}$ ft/min. Rewrite the rate in standard notation.
A dime is the thinnest coin in U.S. currency. A dime’s thickness measures $\text{\hspace{0.17em}}2.2\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}{10}^{6}\text{\hspace{0.17em}}$ m. Rewrite the number in standard notation.
0.00135 m
The average distance between Earth and the Sun is 92,960,000 mi. Rewrite the distance using scientific notation.
A terabyte is made of approximately 1,099,500,000,000 bytes. Rewrite in scientific notation.
$1.0995\times {10}^{12}$
The Gross Domestic Product (GDP) for the United States in the first quarter of 2014 was $\text{\hspace{0.17em}}\text{\$}1.71496\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}{10}^{13}.\text{\hspace{0.17em}}$ Rewrite the GDP in standard notation.
One picometer is approximately $\text{\hspace{0.17em}}3.397\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}{10}^{\mathrm{-11}}\text{\hspace{0.17em}}$ in. Rewrite this length using standard notation.
0.00000000003397 in.
The value of the services sector of the U.S. economy in the first quarter of 2012 was $10,633.6 billion. Rewrite this amount in scientific notation.
For the following exercises, use a graphing calculator to simplify. Round the answers to the nearest hundredth.
${\left(\frac{{12}^{3}{m}^{33}}{{4}^{\mathrm{-3}}}\right)}^{2}$
12,230,590,464 $\text{\hspace{0.17em}}{m}^{66}$
${17}^{3}\xf7{15}^{2}{x}^{3}$
For the following exercises, simplify the given expression. Write answers with positive exponents.
${\left(\frac{{3}^{2}}{{a}^{3}}\right)}^{\mathrm{-2}}{\left(\frac{{a}^{4}}{{2}^{2}}\right)}^{2}$
$\frac{{a}^{14}}{1296}$
${\left({6}^{2}\mathrm{-24}\right)}^{2}\xf7{\left(\frac{x}{y}\right)}^{\mathrm{-5}}$
$\frac{{m}^{2}{n}^{3}}{{a}^{2}{c}^{\mathrm{-3}}}\cdot \frac{{a}^{\mathrm{-7}}{n}^{\mathrm{-2}}}{{m}^{2}{c}^{4}}$
$\frac{n}{{a}^{9}c}$
${\left(\frac{{x}^{6}{y}^{3}}{{x}^{3}{y}^{\mathrm{-3}}}\cdot \frac{{y}^{\mathrm{-7}}}{{x}^{\mathrm{-3}}}\right)}^{10}$
${\left(\frac{{\left(a{b}^{2}c\right)}^{\mathrm{-3}}}{{b}^{\mathrm{-3}}}\right)}^{2}$
$\frac{1}{{a}^{6}{b}^{6}{c}^{6}}$
Avogadro’s constant is used to calculate the number of particles in a mole. A mole is a basic unit in chemistry to measure the amount of a substance. The constant is $\text{\hspace{0.17em}}6.0221413\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}{10}^{23}.\text{\hspace{0.17em}}$ Write Avogadro’s constant in standard notation.
Planck’s constant is an important unit of measure in quantum physics. It describes the relationship between energy and frequency. The constant is written as $\text{\hspace{0.17em}}6.62606957\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}{10}^{\mathrm{-34}}.\text{\hspace{0.17em}}$ Write Planck’s constant in standard notation.
0.000000000000000000000000000000000662606957
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