Definition of Negative and Zero Exponents.
\begin{align*}
a^{-n} \amp = \frac{1}{a^n} \amp\amp (a \ne 0) \\
a^0 \amp = 1 \amp\amp (a \ne 0)
\end{align*}
Section 6.6 Chapter 6 Summary and Review
Subsection 6.6.1 Glossary
- exponent
- power
- power function
- inverse square law
- scientific notation
- root
- radical
- radicand
- index
- irrational number
- principal root
- radical equation
- rationalize
- conjugate
Subsection 6.6.2 Key Concepts
- A positive integer exponent tells us how many copies of its base to multiply together.
- A negative exponent denotes a reciprocal, as long as the base is not zero. A negative exponent does not mean that the power is negative.
Power Function.
A function of the form\begin{equation*} f(x) = kx^p \end{equation*}where \(k\) and \(p\) are nonzero constants, is called a power function.Laws of Exponents.
- \(\displaystyle \displaystyle{a^m\cdot a^n = a^{m+n}}\)
- \(\displaystyle \displaystyle{\frac{a^m}{a^n}=a^{m-n}}\)
- \(\displaystyle \displaystyle{\left(a^m\right)^n=a^{mn}}\)
- \(\displaystyle \displaystyle{\left(ab\right)^n=a^n b^n}\)
- \(\displaystyle \displaystyle{\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}}\)
- The laws of exponents are used to simplify products and quotients of powers, not for sums or differences of powers. We can combine (add or subtract) like terms, but we cannot combine terms with different exponents into a single term.
To Write a Number in Scientific Notation.
- Locate the decimal point so that there is exactly one nonzero digit to its left.
- Count the number of places you moved the decimal point: this determines the power of 10.
- If the original number is greater than 10, the exponent is positive.
- If the original number is less than 1, the exponent is negative.
\(n\)th Roots.
\(s\) is called an \(n\)th root of \(b\) if \(s^n = b\text{.}\)Exponential Notation for Radicals.
For any integer \(n \ge 2\) and for \(a \ge 0\text{,}\)\begin{equation*} a^{1/n} = \sqrt[n]{a} \end{equation*}- It is not possible to write down an exact decimal equivalent for an irrational number, but we can find an approximation to as many decimal places as we like.
- To solve an equation involving \(x^n\text{,}\) we first isolate the power, then raise both sides to the exponent \(\dfrac{1}{n}\text{.}\)
- We can solve an equation where one side is an \(n\)th root of \(x\) by raising both sides of the equation to the \(n\)th power. We must be careful when raising both sides of an equation to an even power, as extraneous solutions may be introduced.
Roots of Real Numbers.
- Every positive number has two real-valued roots, one positive and one negative, if the index is even.
- A negative number has no real-valued root if the index is even.
- Every real number, positive, negative, or zero, has exactly one real-valued root if the index is odd.
- A fractional exponent represents a power and a root. The denominator of the exponent is the root, and the numerator of the exponent is the power. We will define fractional powers only when the base is a positive number.
- Direct variation has the following scaling property: increasing \(x\) by any factor causes \(y\) to increase by the same factor.
Rational Exponents.
\begin{equation*} a^{m/n} = (a^{1/n})^m = (a^m)^{1/n},~~~~a \gt 0\text,~~ n \ne 0 \end{equation*}Rational Exponents and Radicals.
\begin{equation*} a^{m/n} = \sqrt[n]{a^m} =\left( \sqrt[n]{a}\right)^m \end{equation*}- To solve an equation with a fractional exponent we first isolate the power. Then we raise both sides to the reciprocal of the exponent.
Product Rule for Radicals.
\begin{equation*} \displaystyle{\sqrt[n]{ab}=\sqrt[n]{a}\sqrt[n]{b}\text{, }\hphantom{blank000}\text{for } a, b \ge 0} \end{equation*}Quotient Rule for Radicals.
\begin{equation*} \displaystyle{\sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}\text{, }\hphantom{blankblank}\text{for } a\ge 0,~~ b \gt 0} \end{equation*}- In general, it is not true that \(\sqrt[n]{a+b}\) is equivalent to \(\sqrt[n]{a}+\sqrt[n]{b}\text{,}\) or that \(\sqrt[n]{a-b}\) is equivalent to \(\sqrt[n]{a}-\sqrt[n]{b}\text{.}\)
- We simplify radicals by factoring out any perfect powers from the radicand.
- We can add or subtract like radicals in the same way that we add or subtract like terms, by adding or subtracting their coefficients.
- We can use the fundamental principle of fractions to remove radicals from the denominator. This process is called rationalizing the denominator.
- Whenever we raise both sides of an equation to an even power, it is possible to introduce false or extraneous solutions.
Roots of Powers.
\begin{align*} \sqrt[n]{a^n} \amp = a \amp \amp \text{If}~n~\text{is odd.}\\ \sqrt[n]{a^n} \amp = \abs{a} \amp \amp \text{If}~n~\text{is even.} \end{align*}
Exercises 6.6.3 Chapter 6 Review Problems
Exercise Group.
For Problems 1–6, write without negative exponents and simplify.
1.
- \(\displaystyle (-3)^{-4}\)
- \(\displaystyle 4^{-3}\)
Answer.
- \(\displaystyle \dfrac{1}{81}\)
- \(\displaystyle \dfrac{1}{64}\)
2.
- \(\displaystyle (\dfrac{1}{2})^{-2}\)
- \(\displaystyle \dfrac{3}{5^{-2}}\)
Answer.
- \(\displaystyle 9\)
- \(\displaystyle 75\)
3.
- \(\displaystyle (3m)^{-5}\)
- \(\displaystyle -7y{-8}\)
Answer.
- \(\displaystyle \dfrac{1}{243m^5}\)
- \(\displaystyle \dfrac{-7}{y^8}\)
4.
- \(\displaystyle a^{-1}+a^{-2}\)
- \(\displaystyle \dfrac{3q^{-9}}{r^{-2}}\)
Answer.
- \(\displaystyle \dfrac{1}{a}+\dfrac{1}{a^2}\)
- \(\displaystyle \dfrac{3r^2}{q^9}\)
5.
- \(\displaystyle 6c^{-7} \cdot 3^{-1}c^4\)
- \(\displaystyle \dfrac{11z^{-7}}{3^{-2}z^{-5}}\)
Answer.
- \(\displaystyle \dfrac{2}{c^3}\)
- \(\displaystyle \dfrac{99}{z^2}\)
6.
- \(\displaystyle (2d^{-2}k^3)^{-4}\)
- \(\displaystyle \dfrac{2w^3(w^{-2})^{-3}}{5w^{-5}}\)
Answer.
- \(\displaystyle \dfrac{d^8}{16k^{12}}\)
- \(\displaystyle \dfrac{2}{5}w^{14}\)
7.
The speed of light is approximately 186,000 miles per second.
- How long will it take light to travel a distance of 1 foot? (1 mile = 5280 feet) Express your answer in both scientific and standard notation.
- How long does it take sunlight to reach the Earth, a distance of 92,956,000 miles?
Answer.
- \(1.018 \times 10{-9}~\)sec, or 0.000 000 001 018 sec
- 8 min 20 sec
8.
In April 2020, the national debt was over \(24 \times 10^{12}\) dollars. How many hours would it take you to earn an amount equal to the national debt if you were paid $20 per hour? Express your answer in standard notation, both in terms of hours and in terms of years.
Answer.
1,200,000,000,000 hr, or 78,904,109,590 yr
9.
In the twenty-first century, spacecraft may be able to travel at speeds of \(3 \times 10^7\) meters per second, 1000 times their current speed. (At that speed you could circumnavigate the Earth in 1.3 seconds.)
- How long would it take to reach the Sun at this speed? The Sun is approximately \(1.496 \times 10^{11}\) meters from Earth.
- What fraction of the speed of light (\(3 \times 10^8\) meters per second) is this speed?
- How long would it take to reach Proxima Centauri, 4.2 light years from Earth? A light year is the distance that light can travel in one year.
Answer.
- 5000 sec, or 83 min
- \(\displaystyle \dfrac{1}{10}\)
- 42 yr
10.
-
Use the data in the table to calculate the density of each of the planets as follows: first find the volume of the planet, assuming planets are spherical. Then divide the mass of the planet by its volume.
Planet Radius (km) Mass (\(10^{20}\) kg) Density (kg/m\(^3\)) Mercury 2440 3302 Venus 6052 48,690 Earth 6378 59,740 Mars 3397 6419 Jupiter 71,490 18,990,000 Saturn 60,270 5,685,000 Uranus 25,560 866,200 Neptune 24,765 1,028,000 Pluto 1150 150 - The planets are composed of three broad categories of materials: rocky materials, “icy” materials (including water), and the materials that dominate the Sun, namely hydrogen and helium. The density of rock varies from 3000 to 8000 kg/m\(^3\text{.}\) Which of the planets could be composed mainly of rock?
Answer.
Planet Density Mercury 5426 Venus 5244 Earth 5497 Mars 3909 Jupiter 1241 Saturn 620 Uranus 1238 Neptune 1615 Pluto 2355 - Mercury, Venus, Earth, and Mars
Exercise Group.
For Problems 11–14, write each power in radical form.
11.
- \(\displaystyle 25m^{1/2} \)
- \(\displaystyle 8n^{-1/3} \)
Answer.
- \(\displaystyle 25\sqrt{m} \)
- \(\displaystyle \dfrac{8}{\sqrt[3]{n}} \)
12.
- \(\displaystyle (13d)^{2/3} \)
- \(\displaystyle 6x^{2/5}y^{3/5} \)
Answer.
- \(\displaystyle \sqrt[3]{(13d)^3}\)
- \(\displaystyle 6\sqrt[5]{x^2y^3} \)
13.
- \(\displaystyle (3q)^{-3/4} \)
- \(\displaystyle 7(uv)^{3/2} \)
Answer.
- \(\displaystyle \dfrac{1}{\sqrt[4]{27q^3}} \)
- \(\displaystyle 7\sqrt{u^3v^3} \)
14.
- \(\displaystyle (a^2+b^2)^{0.5} \)
- \(\displaystyle (16-x^2)^{0.25} \)
Answer.
- \(\displaystyle \sqrt{a^2+b^2} \)
- \(\displaystyle \sqrt[4]{16-x^2} \)
Exercise Group.
For Problems 15–18, write each radical as a power with a fractional exponent.
15.
- \(\displaystyle 2\sqrt[3]{x^2} \)
- \(\displaystyle \dfrac{1}{4}\sqrt[4]{x} \)
Answer.
- \(\displaystyle 2x^{2/3} \)
- \(\displaystyle \dfrac{1}{4}x^{1/4} \)
16.
- \(\displaystyle z^2\sqrt{z} \)
- \(\displaystyle z\sqrt[3]{z} \)
Answer.
- \(\displaystyle z^{5/2} \)
- \(\displaystyle z^{4/3} \)
17.
- \(\displaystyle \dfrac{6}{\sqrt[4]{b^3}} \)
- \(\displaystyle \dfrac{-1}{3\sqrt[3]{b}} \)
Answer.
- \(\displaystyle 6b^{-3/4} \)
- \(\displaystyle \dfrac{-1}{3}b^{-1/3} \)
18.
- \(\displaystyle \dfrac{-4}{(\sqrt[4]{a})^2} \)
- \(\displaystyle \dfrac{2}{(\sqrt{a})^3} \)
Answer.
- \(\displaystyle -4a^{-1/2} \)
- \(\displaystyle 2a^{-3/2} \)
19.
According to the theory of relativity, the mass of an object traveling at velocity \(v\) is given by the function
\begin{equation*}
m=\dfrac{M}{\sqrt{1-\dfrac{v^2}{c^2}}}
\end{equation*}
where \(M\) is the mass of the object at rest and \(c\) is the speed of light. Find the mass of a man traveling at a velocity of \(0.7c\) if his rest mass is 80 kilograms.
Answer.
112 kg
20.
The cylinder of smallest surface area for a given volume has a radius and height both equal to
\begin{equation*}
\sqrt[3]{\dfrac{V}{\pi}}
\end{equation*}
Find the dimensions of the tin can of smallest surface area with volume 60 cubic inches.
Answer.
Height: 2.673 in; diameter: 5.346 in
21.
Two businesswomen start a small company to produce saddle bags for bicycles. The number of saddle bags, \(q\text{,}\) they can produce depends on the amount of money, \(m\text{,}\) they invest and the number of hours of labor, \(w\text{,}\) they employ, according to the Cobb-Douglas formula
\begin{equation*}
q= 0.6m^{1/4}w^{3/4}
\end{equation*}
where \(m\) is measured in thousands of dollars.
- If the businesswomen invest $100,000 and employ 1600 hours of labor in their first month of production, how many saddle bags can they expect to produce?
- With the same initial investment, how many hours of labor would they need in order to produce 200 saddle bags?
Answer.
- 480
- 498
22.
A child who weighs \(w\) pounds and is \(h\) inches tall has a surface area (in square inches) given approximately by
\begin{equation*}
S= 8.5 h^{0.35}w^{0.55}
\end{equation*}
- What is the surface area of a child who weighs 60 pounds and is 40 inches tall?
- What is the weight of a child who is 50 inches tall and whose surface area is 397 square inches?
Answer.
- 294 sq in
- 90 lb
23.
Membership in the Wildlife Society has grown according to the function
\begin{equation*}
M(t) = 30t^{3/4}
\end{equation*}
where \(t\) is the number of years since its founding in 1970.
- Sketch a graph of the function \(M(t)\text{.}\)
- What was the society’s membership in 1990?
- In what year will the membership be 810 people?
Answer.
- 283
- 2051
24.
The heron population in Saltmarsh Refuge is estimated by conservationists at
\begin{equation*}
P(t) = 360t^{-2/3}
\end{equation*}
where \(t\) is the number of years since the refuge was established in 1990.
- Sketch a graph of the function \(P(t)\text{.}\)
- How many heron were there in 1995?
- In what year will there be only 40 heron left?
25.
A brewery wants to replace its old vats with larger ones. To estimate the cost of the new equipment, the accountant uses the 0.6 rule for industrial costs. This rule states that the cost of a new container is approximately \(~N=Cr^{0.6},~\) where \(C\) is the cost of the old container and \(r\) is the ratio of the capacity of the new container to the old one.
- If an old vat cost $5000, sketch a graph of \(N\) as a function of \(r\) for \(0 \le r \le 5\text{.}\)
- How much should the accountant budget for a new vat that holds 1.8 times as much as the old one?
Answer.
- $7114.32
26.
If a quantity of air expands without changing temperature, its pressure in pounds per square inch is given by \(~P=kV^{-1.4},~\) where \(V\) is the volume of the air in cubic inches and \(k=2.79 \times 10^4\text{.}\)
- Sketch a graph of \(P\) as a function of \(V\) for \(0 \le V \le 100\text{.}\)
- Find the air pressure of an air sample when its volume is 50 cubic inches.
27.
Shipbuilders find that the average cost of producing a ship decreases as more of those ships are produced. This relationship is called the experience curve, and is given by the equation
\begin{equation*}
C = ax^{-b}
\end{equation*}
where \(C\) is the average cost per ship in millions of dollars and \(x\) is the number of ships produced. The value of the constant \(b\) depends on the complexity of the ship. (Source: Storch, Hammon, and Bunch, 1988)
- What is the significance of the constant of proportionality \(a\text{?}\) (Hint: What is the value of \(C\) if only one ship is built?)
- For one kind of ship, \(b = \dfrac{1}{8}\text{,}\) and the cost of producing the first ship is $12 million. Write the equation for \(C\) as a function of \(x\) using radical notation.
- Compute the cost per ship when 2 ships have been built. By what percent does the cost per ship decrease? By what percent does the cost per ship decrease from building 2 ships to building 4 ships?
- By what percent does the average cost decrease from building \(n\) ships to building \(2n\) ships? (In the shipbuilding industry, the average cost per ship usually decreases by 5 to 10% each time the number of ships doubles.)
Answer.
- It is the cost of producing the first ship.
- \(C = \dfrac{12}{ \sqrt[8]{x}} \) million
- About $11 million; about 8.3% ; about 8.3%>
- About 8.3%
28.
A population is in a period of supergrowth if its rate of growth, \(R\text{,}\) at any time is proportional to \(P^k\text{,}\) where \(P\) is the population at that time and \(k\) is a constant greater than \(1\text{.}\) Suppose \(R\) is given by
\begin{equation*}
R = 0.015 P^{1.2}
\end{equation*}
where \(P\) is measured in thousands and \(R\) is measured in thousands per year.
- Find \(R\) when \(P = 20\text{,}\) when \(P = 40\text{,}\) and when \(P = 60\text{.}\)
- What will the population be when its rate of growth is 5000 per year?
- Graph \(R\) and use your graph to verify your answers to parts (a) and (b).
Exercise Group.
In Problems 29 and 30, evaluate the function for the given values.
29.
\(Q(x)=4x^{5/2}\)
- \(\displaystyle Q(16)\)
- \(\displaystyle Q(\dfrac{1}{4})\)
- \(\displaystyle Q(3)\)
- \(\displaystyle Q(100)\)
Answer.
- \(\displaystyle 4096\)
- \(\displaystyle \dfrac{1}{8}\)
- \(\displaystyle 36\sqrt{3} \approx 62.35\)
- \(\displaystyle 400,000\)
30.
\(T(w)=-3w^{2/3}\)
- \(\displaystyle T(27)\)
- \(\displaystyle T(\dfrac{1}{8})\)
- \(\displaystyle T(20)\)
- \(\displaystyle T(1000)\)
Answer.
- \(\displaystyle -27\)
- \(\displaystyle \dfrac{-3}{4}\)
- \(\displaystyle -3\sqrt[3]{400} \approx -22.1\)
- \(\displaystyle -300\)
Exercise Group.
In Problems 31–42, solve.
31.
\(2\sqrt{w}-5=21\)
Answer.
\(169\)
32.
\(16-3\sqrt{w}=-5\)
Answer.
\(49\)
33.
\(12-\sqrt{5v+1}=3\)
Answer.
\(16\)
34.
\(3\sqrt{17-4v}-8=19\)
Answer.
\(-16\)
35.
\(x-3\sqrt{x}+2=0\)
Answer.
\(1,~4\)
36.
\(\sqrt{x+1}+\sqrt{x+8} = 7\)
Answer.
\(8\)
37.
\((x+7)^{1/2} + x^{1/2} = 7\)
Answer.
\(9\)
38.
\((y-3)^{1/2} + (y+4)^{1/2} = 7\)
Answer.
\(12\)
39.
\(\sqrt[3]{x+1}=2\)
Answer.
\(7\)
40.
\(x^{2/3}+2=6\)
Answer.
\(\pm 8\)
41.
\((x-1)^{-3/2} = \dfrac{1}{8}\)
Answer.
\(5\)
42.
\((2x+1)^{-1/2} = \dfrac{1}{3}\)
Answer.
\(4\)
Exercise Group.
For Problems 43–46, solve the formula for the indicated variable.
43.
\(t=\sqrt{\dfrac{2v}{g}}~~~ \text{,}\) for \(g\)
Answer.
\(g=\dfrac{2v}{t^2} \)
44.
\(q-1=2\sqrt{\dfrac{r^2-1}{3}}~~~ \text{,}\) for \(r\)
Answer.
\(r=\dfrac{\pm \sqrt{3q^2-6q+7}}{2} \)
45.
\(R=\dfrac{1+\sqrt{p^2+1}}{2}~~~ \text{,}\) for \(p\)
Answer.
\(p=\pm 2 \sqrt{R^2-R} \)
46.
\(q=\sqrt[3]{\dfrac{1+r^2}{2}}~~~ \text{,}\) for \(r\)
Answer.
\(r=\pm \sqrt{2q^3-1} \)
Exercise Group.
For Problems 47–52, write the radical or radical expression in simplest form.
47.
- \(\displaystyle \sqrt{\dfrac{125p^9}{a^4}}\)
- \(\displaystyle \sqrt[3]{\dfrac{24v^2}{w^6}}\)
Answer.
- \(\displaystyle \dfrac{5p^4}{a^2}\sqrt{5p}\)
- \(\displaystyle \dfrac{2}{w^2}\sqrt[3]{3v^2}\)
48.
- \(\displaystyle \dfrac{\sqrt{a^5b^3}}{\sqrt{ab}}\)
- \(\displaystyle \dfrac{\sqrt{x}\sqrt{xy^3}}{\sqrt{y}}\)
Answer.
- \(\displaystyle a^2b\)
- \(\displaystyle xy\)
49.
- \(\displaystyle \sqrt[3]{8a^3-16b^6}\)
- \(\displaystyle \sqrt[3]{8a^3}\sqrt[3]{-16b^6}\)
Answer.
- \(\displaystyle 2\sqrt[3]{a^3-2b^6}\)
- \(\displaystyle -4ab^2\sqrt[3]{2} \)
50.
- \(\displaystyle \sqrt{4t^2+24t^6}\)
- \(\displaystyle \sqrt{4t^2}\sqrt{24t^6}\)
Answer.
- \(\displaystyle 2t\sqrt{1+6t^4}\)
- \(\displaystyle 4t^4\sqrt{6} \)
51.
- \(\displaystyle (x-2\sqrt{x})^2\)
- \(\displaystyle (x-2\sqrt{x})(x+2\sqrt{x})\)
Answer.
- \(\displaystyle x^2-4x\sqrt{x}+4x\)
- \(\displaystyle x^2-4x\)
52.
- \(\displaystyle (\sqrt{2}-2\sqrt{3})^2\)
- \(\displaystyle (\sqrt{2a}-2\sqrt{b})(\sqrt{2a}+2\sqrt{b})\)
Answer.
- \(\displaystyle 14-4\sqrt{6}\)
- \(\displaystyle 2a-4b\)
Exercise Group.
For Problems 53–56, rationalize the denominator.
53.
- \(\displaystyle \dfrac{7}{\sqrt{5y}}\)
- \(\displaystyle \dfrac{6d}{\sqrt{2d}}\)
Answer.
- \(\displaystyle \dfrac{7\sqrt{5y}}{5y}\)
- \(\displaystyle 3\sqrt{2d}\)
54.
- \(\displaystyle \sqrt{\dfrac{3r}{11s}}\)
- \(\displaystyle \sqrt{\dfrac{26}{2m}}\)
Answer.
- \(\displaystyle \dfrac{\sqrt{33rs}}{11s}\)
- \(\displaystyle \dfrac{\sqrt{13m}}{m}\)
55.
- \(\displaystyle \dfrac{-3}{\sqrt{a}+2}\)
- \(\displaystyle \dfrac{-3}{\sqrt{z}-4}\)
Answer.
- \(\displaystyle \dfrac{-3\sqrt{a}+6}{a-4}\)
- \(\displaystyle \dfrac{-3\sqrt{z}-12}{z-16}\)
56.
- \(\displaystyle \dfrac{2x-\sqrt{3}}{x-\sqrt{3}}\)
- \(\displaystyle \dfrac{m-\sqrt{3}}{5m+2\sqrt{3}}\)
Answer.
- \(\displaystyle \dfrac{2x^2+x\sqrt{3}-3}{x^2-3}\)
- \(\displaystyle \dfrac{5m^2-7m\sqrt{3}+6}{25m^2-12}\)
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