Stewart Precalc 6e Section 1.2: Exponents and Radicals

1 Skill - Scientific Notation in Context · Level 2

Write the number indicated in each statement in scientific notation. *(a)* A light-year is about $5{,}900{,}000{,}000{,}000$ mi. *(b)* The diameter of an electron is about $0.0000000000004$ cm. *(c)* A drop of water contains more than 33 billion billion molecules.

2 Skill - Scientific Notation in Context · Level 2

Write the number indicated in each statement in scientific notation. *(a)* The distance from the earth to the sun is about 93 million miles. *(b)* The mass of an oxygen molecule is about $0.000000000000000000000000053$ kg. *(c)* The mass of the earth is about $5{,}970{,}000{,}000{,}000{,}000{,}000{,}000{,}000$ kg.

3 Skill - Calculations with Scientific Notation · Level 3

Use scientific notation, the Laws of Exponents, and a calculator to perform the indicated operations. State your answer rounded to the number of significant digits indicated by the given data. $(7.2 \times 10^{-9})(1.806 \times 10^{-12})$

4 Skill - Calculations with Scientific Notation · Level 3

Use scientific notation, the Laws of Exponents, and a calculator: $(1.062 \times 10^{24})(8.61 \times 10^{19})$

5 Skill - Calculations with Scientific Notation · Level 3

Use scientific notation, the Laws of Exponents, and a calculator: $\dfrac{1.295643 \times 10^9}{(3.610 \times 10^{-17})(2.511 \times 10^6)}$

6 Skill - Calculations with Scientific Notation · Level 3

Use scientific notation, the Laws of Exponents, and a calculator: $\dfrac{(73.1)(1.6341 \times 10^{28})}{0.0000000019}$

7 Skill - Calculations with Scientific Notation · Level 3

Use scientific notation, the Laws of Exponents, and a calculator: $\dfrac{(0.0000162)(0.01582)}{(594{,}621{,}000)(0.0058)}$

8 Skill - Calculations with Scientific Notation · Level 4

Use scientific notation, the Laws of Exponents, and a calculator: $\dfrac{(3.542 \times 10^{-6})^9}{(5.05 \times 10^4)^{12}}$

9 Skill - Rationalize Denominator · Level 3

Rationalize the denominator. *(a)* $\dfrac{1}{\sqrt{10}}$ *(b)* $\sqrt{\dfrac{2}{x}}$ *(c)* $\sqrt{\dfrac{x}{3}}$

10 Skill - Rationalize Denominator · Level 3

Rationalize the denominator. *(a)* $\sqrt{\dfrac{5}{12}}$ *(b)* $\sqrt{\dfrac{x}{6}}$ *(c)* $\sqrt{y/(2 z)}$

11 Skill - Rationalize Denominator · Level 3

Rationalize the denominator. *(a)* $2/\sqrt[3]{x}$ *(b)* $1/\sqrt[4]{y^3}$ *(c)* $x/y^{\dfrac{2}{5}}$

12 Skill - Rationalize Denominator · Level 3

Rationalize the denominator. *(a)* $1/\sqrt[4]{a}$ *(b)* $a/\sqrt[3]{b^2}$ *(c)* $1/c^{\dfrac{3}{7}}$

13 Skill - Sign Analysis · Level 3

Let $a$, $b$, and $c$ be real numbers with $a > 0$, $b < 0$, and $c < 0$. Determine the sign of each expression. *(a)* $b^5$ *(b)* $b^{10}$ *(c)* $a b^2 c^3$ *(d)* $(b - a)^3$ *(e)* $(b - a)^4$ *(f)* $\dfrac{a^3 c^3}{b^6 c^6}$

14 Theory - Prove Laws of Exponents · Level 3

Prove the given Laws of Exponents for the case in which $m$ and $n$ are positive integers and $m > n$. *(a)* Law 2 *(b)* Law 5 *(c)* Law 6

15 Application - Distance to Star · Level 3

*Distance to the Nearest Star.* Proxima Centauri, the star nearest to our solar system, is 4.3 light-years away. Use the information in Exercise 81(a) (one light-year is about $5.9 \times 10^{12}$ mi) to express this distance in miles.

16 Application - Speed of Light · Level 3

*Speed of Light.* The speed of light is about $186{,}000$ mi/s. Use the information in Exercise 82(a) (distance from sun to earth $\approx 9.3 \times 10^7$ mi) to find how long it takes for a light ray from the sun to reach the earth.

17 Application - Volume of Oceans · Level 3

*Volume of the Oceans.* The average ocean depth is $3.7 \times 10^3$ m, and the area of the oceans is $3.6 \times 10^{14}$ m$^2$. What is the total volume of the ocean in liters? (One cubic meter contains 1000 liters.)

18 Application - National Debt · Level 3

*National Debt.* As of July 2010, the population of the United States was $3.070 \times 10^8$, and the national debt was $1.320 \times 10^{13}$ dollars. How much was each person's share of the debt?

19 Application - Number of Molecules · Level 3

*Number of Molecules.* A sealed room in a hospital, measuring 5 m wide, 10 m long, and 3 m high, is filled with pure oxygen. One cubic meter contains 1000 L, and 22.4 L of any gas contains $6.02 \times 10^{23}$ molecules (Avogadro's number). How many molecules of oxygen are there in the room?

20 Application - Visibility Distance · Level 3

*How Far Can You See?* Because of the curvature of the earth, the maximum distance $D$ that you can see from the top of a tall building of height $h$ is estimated by the formula $D = \sqrt{2 r h + h^2}$ where $r = 3960$ mi is the radius of the earth and $D$ and $h$ are also measured in miles. How far can you see from the observation deck of the Toronto CN Tower, $1135$ ft above the ground?

21 Application - Skidding Car · Level 3

*Speed of a Skidding Car.* Police use the formula $s = \sqrt{30 f d}$ to estimate the speed $s$ (in mi/h) at which a car is traveling if it skids $d$ feet after the brakes are applied suddenly. The number $f$ is the coefficient of friction.

	Tar	Concrete	Gravel
Dry	1.0	0.8	0.2
Wet	0.5	0.4	0.1

*(a)* If a car skids $65$ ft on wet concrete, how fast was it moving when the brakes were applied? *(b)* If a car is traveling at $50$ mi/h, how far will it skid on wet tar?

22 Application - Distance Earth to Sun · Level 4

*Distance from the Earth to the Sun.* It follows from Kepler's Third Law of planetary motion that the average distance from a planet to the sun (in meters) is $d = \left(\dfrac{G M}{4 \pi^2}\right)^{\dfrac{1}{3}} T^{\dfrac{2}{3}}$ where $M = 1.99 \times 10^{30}$ kg is the mass of the sun, $G = 6.67 \times 10^{-11}$ N$\cdot$m$^2$/kg$^2$ is the gravitational constant, and $T$ is the period of the planet's orbit (in seconds). Use the fact that the period of the earth's orbit is about $365.25$ days to find the distance from the earth to the sun.

23 Discovery - Magnitude of a Billion · Level 3

*How Big Is a Billion?* If you had a million ($10^6$) dollars in a suitcase, and you spent a thousand ($10^3$) dollars each day, how many years would it take you to use all the money? Spending at the same rate, how many years would it take you to empty a suitcase filled with a *billion* ($10^9$) dollars?

24 Discovery - Easy Mental Powers · Level 3

*Easy Powers That Look Hard.* Calculate these expressions in your head. Use the Laws of Exponents to help you. *(a)* $18^5/9^5$ *(b)* $20^6 \cdot (0.5)^6$

25 Discovery - Limiting Behavior · Level 4

*Limiting Behavior of Powers.* Complete the following tables. What happens to the $n$th root of $2$ as $n$ gets large? What about the $n$th root of $\dfrac{1}{2}$? Table for $2^{\dfrac{1}{n}}$: $n = 1, 2, 5, 10, 100$. Table for $\left(\dfrac{1}{2}\right)^{\dfrac{1}{n}}$: $n = 1, 2, 5, 10, 100$. Construct a similar table for $n^{\dfrac{1}{n}}$. What happens to the $n$th root of $n$ as $n$ gets large?

26 Discovery - Comparing Roots · Level 3

*Comparing Roots.* Without using a calculator, determine which number is larger in each pair. *(a)* $2^{\dfrac{1}{2}}$ or $2^{\dfrac{1}{3}}$ *(b)* $\left(\dfrac{1}{2}\right)^{\dfrac{1}{2}}$ or $\left(\dfrac{1}{2}\right)^{\dfrac{1}{3}}$ *(c)* $7^{\dfrac{1}{4}}$ or $4^{\dfrac{1}{3}}$ *(d)* $\sqrt[3]{5}$ or $\sqrt{3}$

27 Example - Exponential Notation · Level 2

Evaluate each expression.

(a) $\left(\dfrac{1}{2}\right)^5$

Answer for 27(a)

(b) $(-3)^4$

Answer for 27(b)

(c) $-3^4$

Answer for 27(c)

Enter your answer directly below each part above.

28 Example - Zero and Negative Exponents · Level 2

Evaluate each expression.

(a) $\left(\dfrac{4}{7}\right)^0$

Answer for 28(a)

(b) $x^{-1}$

Answer for 28(b)

(c) $(-2)^{-3}$

Answer for 28(c)

Enter your answer directly below each part above.

29 Example - Using Laws of Exponents · Level 2

Simplify each expression using the Laws of Exponents.

(a) $x^4 x^7$

Answer for 29(a)

(b) $y^4 y^{-7}$

Answer for 29(b)

(c) $\dfrac{c^9}{c^5}$

Answer for 29(c)

(d) $(b^4)^5$

Answer for 29(d)

(e) $(3 x)^3$

Answer for 29(e)

(f) $\left(\dfrac{x}{2}\right)^5$

Answer for 29(f)

Enter your answer directly below each part above.

30 Example - Simplifying Expressions with Exponents · Level 2

Simplify each expression.

(a) $(2 a^3 b^2)(3 a b^4)^3$

Answer for 30(a)

(b) $\left(\dfrac{x}{y}\right)^3 \left(\dfrac{y^2 x}{z}\right)^4$

Answer for 30(b)

Enter your answer directly below each part above.

31 Example - Writing Numbers in Scientific Notation · Level 1

Write each number in scientific notation: (a) \$56,920$ (b) $0.000093$

32 Example - Calculating with Scientific Notation · Level 3

If $a \approx 0.00046$, $b \approx 1.697 \times 10^{22}$, and $c \approx 2.91 \times 10^{-18}$, use a calculator to approximate the quotient $(a b)/c$.

33 Example - Simplifying Expressions Involving nth Roots · Level 2

Simplify each expression: (a) $\sqrt[3]{x^4}$ (b) $\sqrt[4]{81 x^8 y^4}$

34 Example - Combining Radicals · Level 2

Combine like radicals: (a) $\sqrt{32} + \sqrt{200}$ (b) Assume $b > 0$, simplify $\sqrt{25 b} - \sqrt{b^3}$.

35 Example - Using the Definition of Rational Exponents · Level 1

Evaluate or simplify each expression: (a) $4^{\dfrac{1}{2}}$ (b) $8^{\dfrac{2}{3}}$ (c) $125^{-\dfrac{1}{3}}$ (d) $1/\sqrt[3]{x^4}$

36 Example - Using the Laws of Exponents with Rational Exponents · Level 3

Simplify each expression: (a) $a^{\dfrac{1}{3}} a^{\dfrac{7}{3}}$ (b) $\dfrac{a^{\dfrac{2}{5}} a^{\dfrac{7}{5}}}{a^{\dfrac{3}{5}}}$ (c) $(2 a^3 b^4)^{\dfrac{3}{2}}$ (d) $\left(\dfrac{2 x^{\dfrac{3}{4}}}{y^{\dfrac{1}{3}}}\right)^3 \left(\dfrac{y^4}{x^{-\dfrac{1}{2}}}\right)$

37 Example - Simplifying by Writing Radicals as Rational Exponents · Level 2

Simplify each expression: (a) $(2 \sqrt{x})(3 \sqrt[3]{x})$ (b) $\sqrt{x \sqrt{x}}$

38 Example - Rationalizing Denominators · Level 2

Rationalize the denominator: $\dfrac{2}{\sqrt{3}}$

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