Laws of Exponents

5.5. LAWS OF EXPONENTS
303
37. The profit made from selling x wicker baskets is found by subtracting the
costs incurred from the revenue received. In symbols:
P (x) = R(x) − C(x)
Next, replace R(x) and C(x) with their definitions. Because we are supposed
to subtract all of the cost from the revenue, be sure to surround the cost
polynomial with parentheses.
P (x) = 33.45x − (232 + 7x − 0.0085x2)
Distribute the minus sign and combine like terms.
= 33.45x − 232 − 7x + 0.0085x2
= −232 + 26.45x + 0.0085x2
Thus, the profit function is P (x) = −232 + 26.45x + 0.0085x2. Next, to determine the profit if 233 wicker baskets are sold, subsitute 233 for x in the profit
function P (x).
P (x) = −232 + 26.45x + 0.0085x2
P (233) = −232 + 26.45(233) + 0.0085(233)2
You can now use your graphing calculator to determine the profit.
P (233) = 6392.3065
Rounding to the nearest cent, the profit is $6,392.31.
5.5
Laws of Exponents
1. The exponent tells us how many times to write the base as a factor.
(−4)3 = (−4)(−4)(−4)
= −64
Write −4 as a factor 3 times.
Multiply.
3. If you raise any number (other than zero) to the zero power, the answer is
1.
0
5
=1
−
7
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CHAPTER 5. POLYNOMIALS
304
5. The exponent tells us how many times to write the base as a factor.
2 4
4
4
−
= −
−
Write −4/3 as a factor 2 times.
3
3
3
16
Multiply.
=
9
7. If you raise any number (other than zero) to the zero power, the answer is
1.
(−19)0 = 1
9. When multiplying like bases, use the law am an = am+n . That is, repeat
the base and add the exponents.
(7v − 6w)18 · (7v − 6w)17 = (7v − 6w)18+17
35
= (7v − 6w)
Repeat the base, add the exponents.
Simplify: 18 + 17 = 35
11. When multiplying like bases, use the law am an = am+n . That is, repeat
the base and add the exponents.
34 · 30 = 34+0
=3
4
Repeat the base, add the exponents.
Simplify: 4 + 0 = 4
If you wish, you can use your calculator to compute 34 = 81. However, reporting the answer in exponential form 34 is far easier.
13. When multiplying like bases, use the law am an = am+n . That is, repeat
the base and add the exponents.
4n · 48n+3 = 4(n)+(8n+3)
=4
=4
(n+8n)+(3)
9n+3
Repeat the base, add the exponents.
Group like terms.
Simplify.
15. When multiplying like bases, use the law am an = am+n . That is, repeat
the base and add the exponents.
x8 · x3 = x8+3
11
=x
Second Edition: 2012-2013
Repeat the base, add the exponents.
Simplify: 8 + 3 = 11
5.5. LAWS OF EXPONENTS
305
17. When multiplying like bases, use the law am an = am+n . That is, repeat
the base and add the exponents.
25 · 23 = 25+3
=2
Repeat the base, add the exponents.
8
Simplify: 5 + 3 = 8
If you wish, you can use your calculator to compute 28 = 256. However,
reporting the answer in exponential form 28 is far easier.
19. When dividing like bases, use the law am /an = am−n . That is, repeat the
base and subtract the exponents.
416
= 416−16
416
= 40
=1
Repeat the base, subtract the exponents.
Simplify: 16 − 16 = 0
Any number (except zero) to the
zero power equals 1.
Hence, 416 /416 = 1.
21. When dividing like bases, use the law am /an = am−n . That is, repeat the
base and subtract the exponents.
w11
= w11−7
w7
= w4
Repeat the base, subtract the exponents.
Simplify: 11 − 7 = 4
23. When dividing like bases, use the law am /an = am−n . That is, repeat the
base and subtract the exponents.
(9a − 8c)15
= (9a − 8c)15−8
(9a − 8c)8
= (9a − 8c)7
Repeat the base, subtract the exponents.
Simplify: 15 − 8 = 7
25. When dividing like bases, use the law am /an = am−n . That is, repeat the
base and subtract the exponents.
29n+5
= 2(9n+5)−(3n−4)
23n−4
= 29n+5−3n+4
=2
=2
(9n−3n)+(5+4)
6n+9
Repeat the base, subtract the exponents.
Distribute minus sign.
Group like terms.
Simplify.
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306
27. When dividing like bases, use the law am /an = am−n . That is, repeat the
base and subtract the exponents.
417
= 417−9
49
= 48
Repeat the base, subtract the exponents.
Simplify: 17 − 9 = 8
If you wish, you can use your calculator to compute 48 = 65536. However,
reporting the answer in exponential form 48 is far easier.
29. When raising a power to a power, use the law (am )n = amn . That is,
repeat the base and multiply the exponents.
8m−6 7
4
= 47(8m−6)
=4
Repeat the base, multiply the exponents.
56m−42
Distribute 7.
31. When raising a power to a power, use the law (am )n = amn . That is,
repeat the base and multiply the exponents.
7
(9x + 5y)3 = (9x + 5y)(3)(7)
21
= (9x + 5y)
Repeat the base, multiply the exponents.
Simplify: (3)(7) = 21
33. When raising a power to a power, use the law (am )n = amn . That is,
repeat the base and multiply the exponents.
3 2
4
= 4(3)(2)
=4
6
Repeat the base, multiply the exponents.
Simplify: (3)(2) = 6
If you wish, you can use your calculator to compute 46 = 4096. However,
reporting the answer in exponential form 46 is far easier.
35. When raising a power to a power, use the law (am )n = amn . That is,
repeat the base and multiply the exponents.
4 7
c
= c(4)(7)
=c
28
Second Edition: 2012-2013
Repeat the base, multiply the exponents.
Simplify: (4)(7) = 28
5.5. LAWS OF EXPONENTS
307
37. When raising a power to a power, use the law (am )n = amn . That is,
repeat the base and multiply the exponents.
2 0
6
= 6(2)(0)
Repeat the base, multiply the exponents.
= 60
=1
Simplify: (2)(0) = 0
Any number (except zero) raised to
the zero power is 1.
0
Hence, 62 = 1.
39. When raising a product to a power, use the law (ab)n = an bn . That is,
raise each factor to the n.
(uw)5 = u5 w5
Raise each factor to the 5.
41. When raising a product to a power, use the law (ab)n = an bn . That is,
raise each factor to the n.
(−2y)3 = (−2)3 (y)3
= −8y
3
Raise each factor to the 3.
Simplify: (−2)3 = −8
43. When raising a product to a power, use the law (ab)n = an bn . That is,
raise each factor to the n.
(3w9 )4 = (3)4 (w9 )4
= 81w
36
Raise each factor to the 4.
Simplify: (3)4 = 81 and (w9 )4 = w36
45. When raising a product to a power, use the law (ab)n = an bn . That is,
raise each factor to the n.
(−3x8 y 2 )4 = (−3)4 (x8 )4 (y 2 )4
32 8
= 81x y
Raise each factor to the 4.
Simplify: (−3)4 = 81, (x8 )4 = x32 ,
and (y 2 )4 = y 8
47. When raising a product to a power, use the law (ab)n = an bn . That is,
raise each factor to the n.
(7s6n )3 = (7)3 (s6n )3
= 343s
18n
Raise each factor to the 3.
Simplify: (7)3 = 343, (s6n )3 = s18n
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CHAPTER 5. POLYNOMIALS
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49. When raising a quotient to a power, use the law (a/b)n = an /bn . That is,
raise both numerator and denominator to the n.
v 3 v 3
= 3
Raise numerator and denominator
2
2
to the third power.
=
v3
8
Simplify: (2)3 = 8
51. When raising a quotient to a power, use the law (a/b)n = an /bn . That is,
raise both numerator and denominator to the n. When you raise a negative
fraction to an even power, the answer is positive.
2
22
2
= 2
Raise numerator and denominator
−
u
u
to the second power.
4
Simplify: (2)2 = 4
= 2
u
53. When raising a quotient to a power, use the law (a/b)n = an /bn . That is,
raise both numerator and denominator to the n. When you raise a negative
fraction to an even power, the answer is positive.
8 4
8 4
r
r
=
Raise numerator and denominator
−
5
54
to the fourth power.
4
r32
=
Simplify: r8 = r32 and (5)4 = 625
625
55. When raising a quotient to a power, use the law (a/b)n = an /bn . That is,
raise both numerator and denominator to the n.
4
54
5
=
Raise numerator and denominator
4
c9
(c9 )
to the fourth power.
4
625
Simplify: (5)4 = 625 and c9 = c36
= 36
c
57. When raising a quotient to a power, use the law (a/b)n = an /bn . That is,
raise both numerator and denominator to the n.
2
5
52
=
Raise numerator and denominator
12
u
(u12 )2
to the second power.
2
25
= 24
Simplify: (5)2 = 25 and u12 = u24
u
Second Edition: 2012-2013