Section 1
Exponents and Scientific Notation
Write each expression with positive exponents. See Examples 5 and 6.$5 x^{-1} y^{-2}$
Write each expression with positive exponents. See Examples 5 and 6. $7 x y^{-4}$
Write each expression with positive exponents. See Examples 5 and 6.$a^2 b^{-1} c^{-5}$
Write each expression with positive exponents. See Examples 5 and 6.$a^{-4} b^2 c^{-6}$
Write each expression with positive exponents. See Examples 5 and 6.$\frac{y^{-2}}{x^{-4}}$
Write each expression with positive exponents. See Examples 5 and 6.$\frac{x^{-7}}{z^{-3}}$
Use the product rule to simplify each expression. See Examples 1 and 2.$4^2 \cdot 4^3$
Use the product rule to simplify each expression. See Examples 1 and 2. $3^3 \cdot 3^5$
Use the product rule to simplify each expression. See Examples 1 and 2.$x^5 \cdot x^3$
Use the product rule to simplify each expression. See Examples 1 and 2.$a^2 \cdot a^9$
Use the product rule to simplify each expression. See Examples 1 and 2.$m \cdot m^7 \cdot m^6$
Use the product rule to simplify each expression. See Examples 1 and 2.$n \cdot n^{10} \cdot n^{12}$
Use the product rule to simplify each expression. See Examples 1 and 2.$(4 x y)(-5 x)$
Use the product rule to simplify each expression. See Examples 1 and 2.$(-7 x y)(7 y)$
Use the product rule to simplify each expression. See Examples 1 and 2. $\left(-4 x^3 p^2\right)\left(4 y^3 x^3\right)$
Use the product rule to simplify each expression. See Examples 1 and 2.$\left(-6 a^2 b^3\right)\left(-3 a b^3\right)$
Evaluate each expression. See Example 3.$-8^0$
Evaluate each expression. See Example 3.$(-9)^0$
Evaluate each expression. See Example 3.$(4 x+5)^0$
Evaluate each expression. See Example 3. $(3 x-1)^0$
Evaluate each expression. See Example 3.$-x^0$
Evaluate each expression. See Example 3.$-5 x^0$
Evaluate each expression. See Example 3.$4 x^0+5$
Evaluate each expression. See Example 3.$8 x^0+1$
Use the quotient rule to simplify. See Example 4.$\frac{a^5}{a^2}$
Use the quotient rule to simplify. See Example 4.$\frac{x^9}{x^4}$
Use the quotient rule to simplify. See Example 4. $-\frac{26 z^{11}}{2 z^7}$
Use the quotient rule to simplify. See Example 4.$-\frac{16 x^5}{8 x}$
Use the quotient rule to simplify. See Example 4.$\frac{x^9 y^6}{x^8 y^6}$
Use the quotient rule to simplify. See Example 4. $\frac{a^{12} b^2}{a^9 b}$
Use the quotient rule to simplify. See Example 4.$\frac{12 x^4 y^7}{9 x y^5}$
Use the quotient rule to simplify. See Example 4.$\frac{24 a^{10} b^{11}}{10 a b^3}$
Use the quotient rule to simplify. See Example 4.$\frac{-36 a^5 b^7 c^{10}}{6 a b^3 c^4}$
Use the quotient rule to simplify. See Example 4.$\frac{49 a^3 b c^{14}}{-7 a b c^8}$
Simplify and write using positive exponents only. See Examples 5 and 6. $4^{-2}$
Simplify and write using positive exponents only. See Examples 5 and 6.$2^{-3}$
Simplify and write using positive exponents only. See Examples 5 and 6.$(-3)^{-3}$
Simplify and write using positive exponents only. See Examples 5 and 6. $(-6)^{-2}$
Simplify and write using positive exponents only. See Examples 5 and 6.$\frac{x^7}{x^{15}}$
Simplify and write using positive exponents only. See Examples 5 and 6.$\frac{z}{z^3}$
Simplify and write using positive exponents only. See Examples 5 and 6.$5 a^{-4}$
Simplify and write using positive exponents only. See Examples 5 and 6.$10 b^{-1}$
Simplify and write using positive exponents only. See Examples 5 and 6.$\frac{x^{-7}}{y^{-2}}$
Simplify and write using positive exponents only. See Examples 5 and 6. $\frac{p^{-13}}{q^{-3}}$
Simplify and write using positive exponents only. See Examples 5 and 6. $\frac{x^{-2}}{x^5}$
Simplify and write using positive exponents only. See Examples 5 and 6.$\frac{z^{-12}}{z^{10}}$
Simplify and write using positive exponents only. See Examples 5 and 6.$\frac{8 r^4}{2 r^{-4}}$
Simplify and write using positive exponents only. See Examples 5 and 6.$\frac{3 s^3}{15 s^{-3}}$
Simplify and write using positive exponents only. See Examples 5 and 6.$\frac{x^{-9} x^4}{x^{-5}}$
Simplify and write using positive exponents only. See Examples 5 and 6.$\frac{y^{-7} y}{y^8}$
Simplify and write using positive exponents only. See Examples 5 and 6.$\frac{2 a^{-6} b^2}{18 a b^{-5}}$
Simplify and write using positive exponents only. See Examples 5 and 6.$\frac{18 a b^{-6}}{3 a^{-3} b^6}$
Simplify and write using positive exponents only. See Examples 5 and 6. $\frac{\left(24 x^8\right)(x)}{20 x^{-7}}$
Simplify and write using positive exponents only. See Examples 5 and 6.$\frac{\left(30 z^2\right)\left(z^5\right)}{55 z^{-4}}$
Simplify and write using positive exponents only. See Examples 1 through 6. $-7 x^3 \cdot 20 x^9$
Simplify and write using positive exponents only. See Examples 1 through 6.$-3 y \cdot-9 y^4$
Simplify and write using positive exponents only. See Examples 1 through 6. $x^7 \cdot x^8 \cdot x$
Simplify and write using positive exponents only. See Examples 1 through 6.$y^6 \cdot y \cdot y^9$
Simplify and write using positive exponents only. See Examples 1 through 6.$2 x^3 \cdot 5 x^7$
Simplify and write using positive exponents only. See Examples 1 through 6.$-3 z^4 \cdot 10 z^7$
Simplify and write using positive exponents only. See Examples 1 through 6.$(5 x)^0+5 x^0$
Simplify and write using positive exponents only. See Examples 1 through 6. $4 y^0-(4 y)^0$
Simplify and write using positive exponents only. See Examples 1 through 6.$\frac{z^{12}}{z^{15}}$
Simplify and write using positive exponents only. See Examples 1 through 6.$\frac{x^{11}}{x^{20}}$
Simplify and write using positive exponents only. See Examples 1 through 6. $3^0-3 t^0$
Simplify and write using positive exponents only. See Examples 1 through 6.$4^0+4 x^0$
Simplify and write using positive exponents only. See Examples 1 through 6. $\frac{y^{-3}}{y^{-7}}$
Simplify and write using positive exponents only. See Examples 1 through 6.$\frac{y^{-6}}{y^{-9}}$
Simplify and write using positive exponents only. See Examples 1 through 6.$4^{-1}+3^{-2}$
Simplify and write using positive exponents only. See Examples 1 through 6.$1^{-3}-4^{-2}$
Simplify and write using positive exponents only. See Examples 1 through 6. $3 x^{-1}$
Simplify and write using positive exponents only. See Examples 1 through 6.$(4 x)^{-1}$
Simplify and write using positive exponents only. See Examples 1 through 6.$\frac{r^4}{r^{-4}}$
Simplify and write using positive exponents only. See Examples 1 through 6.$\frac{x^{-5}}{x^3}$
Simplify and write using positive exponents only. See Examples 1 through 6.$\frac{x^{-7} y^{-2}}{x^2 y^2}$
Simplify and write using positive exponents only. See Examples 1 through 6.$\frac{a^{-5} b^7}{a^{-2} b^{-3}}$
Simplify and write using positive exponents only. See Examples 1 through 6.$\left(-4 x^2 y\right)\left(3 x^4\right)\left(-2 x y^5\right)$
Simplify and write using positive exponents only. See Examples 1 through 6.$\left(-6 a^4 b\right)\left(2 b^3\right)\left(-3 a b^6\right)$
Simplify and write using positive exponents only. See Examples 1 through 6.$2^{-4} \cdot x$
Simplify and write using positive exponents only. See Examples 1 through 6.$5^{-2} \cdot y$
Simplify and write using positive exponents only. See Examples 1 through 6.$\frac{5^{17}}{5^{13}}$
Simplify and write using positive exponents only. See Examples 1 through 6.$\frac{10^{25}}{10^{23}}$
Simplify and write using positive exponents only. See Examples 1 through 6.$\frac{8^{-7}}{8^{-6}}$
Simplify and write using positive exponents only. See Examples 1 through 6.$\frac{13^{-10}}{13^{-9}}$
Simplify and write using positive exponents only. See Examples 1 through 6.. $\frac{9^{-5} a^4}{9^{-3} a^{-1}}$
Simplify and write using positive exponents only. See Examples 1 through 6. $\frac{11^{-9} b^3}{11^{-7} b^{-4}}$
Simplify and write using positive exponents only. See Examples 1 through 6.. $\frac{14 x^{-2} y z^{-4}}{2 x y z}$
Simplify and write using positive exponents only. See Examples 1 through 6. $\frac{30 x^{-7} y z^{-14}}{3 x y z}$
Simplify. Assume that variables in the exponents represent nonzero integers and that $x, y$, and $z$ are not 0 . See Example 7$x^5 \cdot x^{7 a}$
Simplify. Assume that variables in the exponents represent nonzero integers and that $x, y$, and $z$ are not 0 . See Example 7$x^{4 a} \cdot x^7$
Simplify. Assume that variables in the exponents represent nonzero integers and that $x, y$, and $z$ are not 0 . See Example 7$\frac{x^{3 /-1}}{x^5}$
Simplify. Assume that variables in the exponents represent nonzero integers and that $x, y$, and $z$ are not 0 . See Example 7$\frac{y^{4 p-2}}{y^{3 p}}$
Simplify. Assume that variables in the exponents represent nonzero integers and that $x, y$, and $z$ are not 0 . See Example 7$y^{2 p} \cdot y^{9 p}$
Simplify. Assume that variables in the exponents represent nonzero integers and that $x, y$, and $z$ are not 0 . See Example 7$x^{9 y} \cdot x^{-7 y}$
Simplify. Assume that variables in the exponents represent nonzero integers and that $x, y$, and $z$ are not 0 . See Example 7$\frac{z^{6 x}}{z^7}$
Simplify. Assume that variables in the exponents represent nonzero integers and that $x, y$, and $z$ are not 0 . See Example 7$\frac{y^6}{y^{47}}$
Simplify. Assume that variables in the exponents represent nonzero integers and that $x, y$, and $z$ are not 0 . See Example 7$\frac{x^{3 t} \cdot x^{4 t-1}}{x^4}$
Simplify. Assume that variables in the exponents represent nonzero integers and that $x, y$, and $z$ are not 0 . See Example 7$\frac{z^{5 x} \cdot z^{x-7}}{z^x}$
Write each number in scientific notation. See Example 8.31,250,000
Write each number in scientific notation. See Example 8.. 678,000
Write each number in scientific notation. See Example 8.. 0.016
Write each number in scientific notation. See Example 8.0,007613
Write each number in scientific notation. See Example 8.67,413
Write each number in scientific notation. See Example 8.$36,800,000$
Write each number in scientific notation. See Example 8. 0.0125
Write each number in scientific notation. See Example 8.0.00084
Write each number in scientific notation. See Example 8.0.000053
Write each number in scientific notation. See Example 8.
$98,700,000,000$
Write each number in scientific notation.The approximate distance between Jupiter and the sun is 778,300,000 kilometers. (Source: National Space Data Center)
Write each number in scientific notation.
For the 2010-2011 National Football League season, the payroll of the Super Bowl champion Green Bay Packers was approximately \$72,245,500. (Source: USA Today)
The gross domestic product (GDP) of a region is the value of all final goods and services produced within a given year. The GDP of the European Union for 2010 was estimated to be equivalent to U.S. $\$ 16,228,000,000,000$. (Source: Wikipedia)
By the middle of 2012 , Nintendo had sold $96,560,000$ Wii units. (Source: Nintendo)
Lake Mead, created from the Colorado River by the Hoover Dam, has a capacity of $124,000,000,000$ cubic feet of water. (Source: U.S. Bureau of Reclamation)
The temperature of the core of the sun is about $27,000,000^{\circ} \mathrm{F}$.
A pulsar is a rotating neutron star that gives off sharp, regular pulses of radio waves. For one particular pulsar, the rate of pulses is every 0.001 second.
To convert from cubic inches to cubic meters, multiply by 0.0000164 .
Write each number in standard notation, without exponents. See Example 9.$3.6 \times 10^{-9}$
Write each number in standard notation, without exponents. See Example 9. $2.7 \times 10^{-5}$
Write each number in standard notation, without exponents. See Example 9. $9.3 \times 10^7$
Write each number in standard notation, without exponents. See Example 9. $6.378 \times 10^8$
Write each number in standard notation, without exponents. See Example 9.$1.278 \times 10^6$
Write each number in standard notation, without exponents. See Example 9.$7.6 \times 10^4$
Write each number in standard notation, without exponents. See Example 9.$7.35 \times 10^{12}$
Write each number in standard notation, without exponents. See Example 9.$1.66 \times 10^{-5}$
Write each number in standard notation, without exponents. See Example 9.$4.03 \times 10^{-7}$
Write each number in standard notation, without exponents. See Example 9.$8.007 \times 10^8$
Write each number in standard notation.The estimated world population in 1 C.E. was $3.0 \times 10^8$. (Source: World Almanac and Book of Facts)
Write each number in standard notation.There are $3.949 \times 10^6$ miles of highways, roads, and streets in the United States. (Source: Bureau of Transportation Statistics)
Write each number in standard notation.Chiricahua National Monument in Arizona contains $9.5 \times 10^{-3}$ square kilometers of privately owned land.
Write each number in standard notation.Hagerman Fossil Beds National Monument in Idaho contains $6.68 \times 10^{-2}$ square kilometers of privately owned land.
Write each number in standard notation.
In 2010, the Coca-Cola Company sold $2.55 \times 10^{10}$ unit cases of beverages worldwide. (Source: the Coca-Cola Company)
In 2010, the revenue for McDonald's was $\$ 2.27 \times 10^{10}$. (Source: McDonald's Corporation)
Evaluate. See Sections 1.3 and 5.1.$(5 \cdot 2)^2$
Evaluate. See Sections 1.3 and 5.1.$5^2 \cdot 2^2$
Evaluate. See Sections 1.3 and 5.1. $\left(\frac{3}{4}\right)^3$
Evaluate. See Sections 1.3 and 5.1.$\frac{3^3}{4^3}$
Evaluate. See Sections 1.3 and 5.1.$\left(2^3\right)^2$
Evaluate. See Sections 1.3 and 5.1. $\left(2^2\right)^3$
Explain how to convert a number from standard notation to scientific notation.
Explain how to convert a number from scientific notation to standard notation.
Explain why $(-5)^0$ simplifies to 1 but $-5^0$ simplifies to -1 .
Explain why both $4 x^0-3 y^0$ and $(4 x-3 y)^0$ simplify to 1 .
Simplify where possible.a. $x^e \cdot x^e$b. $x^n+x^n$c. $\frac{x^a}{x^b}$d. $x^a \cdot x^b$e. $x^a+x^b$
Which numbers are equal to 36,000 ? Of these, which is written in scientific notation?a. $36 \times 10^3$b. $360 \times 10^2$c. $0.36 \times 10^5$d. $3.6 \times 10^4$
Without calculating, determine which number is larger.$7^{11}$ or $7^{13}$
Without calculating, determine which number is larger.$5^{10}$ or $5^9$
Without calculating, determine which number is larger.$7^{-11}$ or $7^{-13}$
Without calculating, determine which number is larger.$5^{-10}$ or $5^{-9}$