1 Properties of opposites (Section P.1 , page 13)
2 Variables (Section P.1 , page 2)
In 2012, Americans drank about 110 billion cups of coffee and spent more than $29 billion on candy and other confectionery products. The American population at that time was about 314 million. Because our day-to-day activities bring us into contact with more manageable quantities, most people find large numbers like those just cited a bit difficult to understand. Using exponents, the method of scientific notation allows us to write large and small quantities in a manner that makes comparing such quantities fairly easy. In turn, these comparisons help us get a better perspective on these quantities. In Example 9 of this section, without relying on a calculator, we see that if we distribute the cost of the candy and other confectionery products used in 2012 evenly among all individuals in the United States, each person will spend over $92. Distributing the coffee used in 2012 evenly among all individuals in the United States gives each person about 350 cups of coffee.
1 Use integer exponents.
In Section P.1, we introduced the following notation:
If a is a real number and n is a positive integer, then
We now define an
Negative exponents indicate the reciprocal of a number. Zero cannot be used as a base with a negative exponent because zero does not have a reciprocal. Furthermore, 00
Evaluate.
(−5)−2
−5−2
80
(23)−3
(−5)−2=1(−5)2=125The exponent −2 applies to the base −5.
−5−2=−152=−125The exponent −2 applies to the base 5.
80=1By definition
(23)−3=1(23)3=1827=278(23)3=23⋅23⋅23=827
Evaluate.
2−1
(45)0
(32)−2
In Example 1, we see that parts a and b give different results. In part a, the base is −5
2 Use the rules of exponents.
We now review the rules of exponents.
Let’s see what happens when we multiply a5
Similarly, if you multiply m factors of a by n factors of a, you get m+n
Simplify. Use the product rule and (if necessary) the definition of a negative exponent or reciprocal to write each answer without negative exponents.
2x3⋅x5
x7⋅x−7
(−4y2)(3y7)
2x3⋅x5=2x3+5=2x8Add exponents: 3+5=8.
x7⋅x−7=x7+(−7)=x0=1Add exponents: 7+(−7)=0; simplify.
(−4y2)(3y7)=(−4)3y2y7Group the factors with variable bases.=−12y2+7Add exponents.=−12y92+7=9
Simplify. Write each answer without using negative exponents.
x2⋅3x7
(22x3)(4x−3)
Look what happens when we divide a5
So a5a3=a5⋅1a3=a5⋅a−3=a5−3=a2.
Remember that if a≠0,
Simplify. Use the quotient rule to write each answer without negative exponents.
510510
2−123
x−3x5
510510=510−10=50=1
2−123=2−1−3=2−4=124=116
x−3x5=x−3−5=x−8=1x8
Simplify. Write each answer without negative exponents.
3430
55−2
2x33x−4
To introduce the next rule of exponents, we consider (23)4.
So (23)4=23⋅4=212.
This suggests the following rule.
Simplify. Write each answer without negative exponents.
(52)0
[(−3)2]3
(x3)−1
(x−2)−3
(52)0=52⋅0=50=1
[(−3)2]3=(−3)2⋅3=(−3)6=729For (−3)6, the base is −3.
(x3)−1=x3(−1)=x−3=1x3
(x−2)−3=x(−2)(−3)=x6
Simplify. Write each answer without negative exponents.
(7−5)0
(70)−5
(x−1)8
(x−2)−5
We now consider the power of a product.
So (2⋅3)5=25⋅35.
This suggests the following rule.
Simplify. Use the power-of-a-product rule to write each expression without negative exponents.
(3x)2
(−3x)−2
(−32)3
(xy)−4
(x2y)3
(3x)2=32x2=9x2
(−3x)−2=1(−3x)2=1(−3)2x2=19x2Negative exponents denote reciprocals.
(−32)3=(−1⋅32)3=(−1)3(32)3=(−1)(36)=−729
(xy)−4=1(xy)4=1x4y4Negative exponents denote reciprocals.
(x2y)3=(x2)3y3=x2⋅3y3=x6y3Recall that (x2)3=x2⋅3.
Simplify. Write each answer without negative exponents.
(12x)−1
(5x−1)2
(xy2)3
(x−2y)−3
To see the last rule, we consider (32)5.
More generally, we have the following rules.
Simplify. Use the power-of-quotient rules to write each answer without negative exponents.
(35)3
(23)−2
(35)3=3353=27125Equation (2)
(23)−2=(32)2=3222=94Equation (3) followed by equation (2)
Simplify. Write each answer without negative exponents.
(13)2
(107)−2
3 Simplify exponential expressions.
There are many correct ways to simplify exponential expressions. The order in which you apply the rules for exponents is a matter of personal preference.
Simplify the following.
(−4x2y3)(7x3y)
(x52y−3)−3
(−4x2y3)(7x3y)=(−4)(7)x2x3y3yGroup factors with the same base.=−28x2+3y3+1Apply the product rule to add exponents;remember that y=y1.=−28x5y4
(x52y−3)−3=(x5)−3(2y−3)−3The exponent −3 is applied to both the numeratorand the denominator.=x5(−3)2−3(y−3)−3Multiply exponents in the numerator; apply theexponent −3 to each factor in the denominator.=x−152−3y(−3)(−3)Power rule for exponents; 5(−3)=−15.=x−152−3y9(−3)(−3)=9=x−15x1523x15232−3y9Multiply numerator and denominator by x1523.=23x15y9x−15x15=x0=1; 232−3=20=1=8x15y923=8
Simplify each expression.
(2x4)−2
x2(−y)3(xy2)3
4 Use scientific notation.
Scientific measurements and calculations often involve very large or very small positive numbers. For example, 1 gram of oxygen contains approximately
and the mass of one oxygen atom is approximately
Such numbers contain so many zeros that they are awkward to work with in calculations. Fortunately, scientific notation provides a better way to write and work with such large and small numbers.
Scientific notation consists of the product of a number less than 10, and greater than or equal to 1, and an integer power of 10. That is, scientific notation of a number has the form
where c is a real number in decimal notation with 1≤c<10
Write each decimal number in scientific notation.
421,000
10
3.621
0.000561
Because 421,000=421000.0,
Because the decimal point is moved five places to the left, the exponent is positive and we write
The decimal point for 10 is to the right of the units digit. Count one place to move the decimal point between 1 and 0 and produce the number 1.0.
Because the decimal point is moved one place to the left, the exponent is positive and we write
The number 3.621 is already between 1 and 10, so the decimal does not need to be moved. We write
The decimal point in 0.000561 must be moved between the 5 and the 6 to produce the number 5.61. We count four places as follows:
Because the decimal point is moved four places to the right, the exponent is negative and we write
Write 732,000 in scientific notation.
Note that in scientific notation, “small” numbers (positive numbers less than 1) have negative exponents and “large” numbers (numbers greater than 10) have positive exponents.
At the beginning of this section, we mentioned that in 2012, Americans drank about 110 billion cups of coffee and spent more than $29 billion on candy and other confectionery products. To see how these products would be evenly distributed among the population, we first convert those numbers to scientific notation.
The U.S. population in 2012 was about 314 million, and 314 million is 314,000,000=3.14×108.
To distribute the coffee evenly among the population, we divide:
To distribute the cost of the candy evenly among the population, we divide:
or about $92 per person.
If the amount spent on candy consumption remains unchanged when the U.S. population reaches 325 million, what is the cost per person when cost is distributed evenly throughout the population?
In the expression 7−2,
In the expression −37,
The number 14−2
The power-of-a-product rule allows us to rewrite (5a)3
True or False. (−11)10=−1110.
True or False. The exponential expression (x2)3
True or False. (ab)n=anbn
True or False. (a+b)n=an+bn
In Exercises 9–46, evaluate each expression.
3−2
2−3
(12)−4
(−12)−2
70
(−8)0
−(−7)0
(√2)0
(23)2
(32)3
(32)−2
(72)−1
(5−2)3
(5−1)3
(4−3)⋅(45)
(7−2)⋅(73)
(√3)0+100
(√5)0−90
3−2+(13)2
5−2+(15)2
−2−3
−3−2
(−3)−2
(−2)−3
211210
3638
(53)4512
(95)298
25⋅3−224⋅3−3
4−2⋅534−3⋅5
−5−22−1
−7−23−1
(23)−1
(15)−1
(23)−2
(32)−2
(117)−2
(135)−2
In Exercises 47–86, simplify each expression. Write your answers without negative exponents. Whenever an exponent is negative or zero, assume that the base is not zero.
x4y0
x−1y0
x−1y
x2y−2
x−1y−2
x−3y−2
(x−3)4
(x−5)2
(x−11)−3
(x−4)−12
−3(xy)5
−8(xy)6
4(xy−1)2
6(x−1y)3
3(x−1y)−5
−5(xy−1)−6
(x3)2(x2)5
x2(x3)4
(2xyx2)3
(5xyx3)4
(−3x2yx)5
(−2xy2y)3
(−3x5)−2
(−5y3)−4
(4x−2xy5)3
(3x2yy3)5
x3y−3x−2y
x2y−2x−1y2
27x−3y59x−4y7
15x5y−23x7y−3
1x3(x2)3x−4
(8a3b)−4(2ab)12
(−xy2)3(−2x2y2)−4
[(−x2 y)3y−4(xy)5]−2
(4x3y2z)2(x3y2z)−7
(2xyz)2(x3y)2(xz)−1
5a−2bc2a4b−3c2
(−3)2a5(bc)2(−2)3a2b3c4
(xy−3z−2x2y−4z3)−3
(xy−2z−1x−5yz−8)−1
In Exercises 87–94, write each number in scientific notation.
125
247
850,000
205,000
0.007
0.0019
0.00000275
0.0000038
In Exercises 95 and 96, use the fact that when you uniformly stretch or shrink a three-dimensional object in every direction by a factor of a, the volume of the resulting figure is scaled by a factor of a3. For example, when you uniformly scale (stretch or shrink) a three-dimensional object by a factor of 2, the volume of the resulting figure is scaled by a factor of 23, or 8.
A display in the shape of a baseball bat has a volume of 135 cubic feet. Find the volume of the display that results by uniformly scaling the figure by a factor of 2.
A display in the shape of a football has a volume of 675 cubic inches. Find the volume of the display that results by uniformly scaling the figure by a factor of 13.
The area A of a square with side of length x is given by A=x2. Use this relationship to
verify that doubling the length of the side of a square floor increases the area of the floor by a factor of 22.
verify that tripling the length of the side of a square floor increases the area of the floor by a factor of 32.
The area A of a circle with diameter d is given by A=π(d2)2. Use this relationship to
verify that doubling the length of the diameter of a circular skating rink increases the area of the rink by a factor of 22.
verify that tripling the length of the diameter of a circular skating rink increases the area of the rink by a factor of 32.
The cross section of one type of weight-bearing rod (the surface you would get if you sliced the rod perpendicular to its axis) used in the Olympics is a square, as shown in the accompanying figure. The rod can handle a stress of 25,000 pounds per square inch (psi). The relationship between the stress S the rod can handle, the load F the rod can carry, and the width w of the cross section is given by the equation Sw2=F. Assuming that w=0.25 inch, find the load the rod can support.
The cross section of one type of weight-bearing rod (the surface you would get if you sliced the rod perpendicular to its axis) used in the Olympics is a circle, as shown in the figure. The rod can handle a stress of 10,000 pounds per square inch (psi). The relationship between the stress S the rod can handle, the load F the rod can carry, and the diameter d of the cross section is given by the equation Sπ(d/2)2=F. Assuming that d=1.5 inches, find the load the rod can support. Use π≈3.14 in your calculation.
A year has 365.25 days. Write the number of seconds in one year in scientific notation.
Repeat Exercise 101 for a leap year (366 days).
Complete the following table.
Celestial Body | Equatorial Diameter (km) | Scientific Notation |
---|---|---|
Earth | 12,700 | |
Moon | 3.48×103 (km) | |
Sun | 1,390,000 | |
Jupiter | 1.34×105 (km) | |
Mercury | 4800 |
In Exercises 104–110, express the number in each statement in scientific notation.
One gram of oxygen contains about
One gram of hydrogen contains about
One oxygen atom weighs about
One hydrogen atom weighs about
The distance from Earth to the moon is about
The mass of Earth is about
The mass of the sun is about
If 2x=32, find
2x+2
2x−1
If 3x=81, find
3x+1
3x−2
If 5x=11, find
5x+1
5x−2
If ax=b, find
ax+2
ax−1
Simplify: 32−n⋅92n−233n
Simplify: 2m+1⋅32m−n⋅5m+n+2⋅6n6m⋅10n+1⋅15m
[Hint: 6=2⋅3, 10=2⋅5, 15=3⋅5]
Simplify: 2x(y−z)2y(x−z)÷(2y2x)z
Simplify: (axay)1xy⋅(ayaz)1yz⋅(azax)1xz
Simplify:
x2⋅x5
(2x)(−5x2)
(2y2)(3y3)(4y5)
Simplify:
2x2+5x2
3x2−4x2
3x3−5x3+11x3
True or False: 5x2+3x3=8x5
Use the distributive property to simplify: 2x2(5x3−3x+4)