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13

# EXPONENTS

Three rules

WHEN All THE FACTORS OF A PRODUCT are equal -- 6· 6· 6· 6 -- we call the product a power of that factor (Lesson 1).

Do the problem yourself first!

Problem 1.   What number is

 a) the third power of 2?  2· 2· 2 = 8 b) the fourth power of 3? = 81 c) the fifth power of 10?  = 100,000 d) the first power of 8? = 8

Now, rather than write the third power of 2 as 2· 2· 2, we write 2 just once -- and place an exponent:  23.   2 is called the base.  The exponent indicates the number of times to repeat the base as a factor.

The student must take care not to confuse 3a, which means 3 times a, with a3, which means a times a times a.

 3a = a + a + a, a3 = a· a· a.

Problem 2.   What does each symbol mean?

 a) x5 = xxxxx b) 53 = 5· 5· 5 c) 5· 3 = 3 + 3 + 3 + 3 + 3. d) (5a)3 = 5a· 5a· 5a e) 5a3 = 5aaa

In part d), the parentheses indicate that the base is 5a.  In part e), only a is the base.  The exponent does not apply to 5.

Problem 3.   34 = 81.

a)   Which number is called the base?   3

b)   Which number is the power?   81 is the power of 3.

c)   Which number is the exponent?

4.  It indicates the power, namely the 4th.

Problem 4.   Write out the meaning of these symbols.

 a) a2a3 = aa· aaa b) (ab)3 = ab· ab· ab c) (a2)3 = a2· a2· a2

Problem 5.   Write out the meaning of these symbols.  In each one, what is the base?

a)   a4aaaa.  The base is a.

 b) −a4 = −aaaa.  The base again is a. This is the negative of a4.A minus sign always signifies the negative of the number that follows. −5 is the negative of 5. And −a4 is the negative of a4.

c)   (−a)4 = (−a)(−a)(−a)(−a).  Here, the base is (−a).

Problem 6.   Evaluate.

a)  24 = 16.

b)  −24 = −16.  This is the negative of 24. The base is 2. See
Problem 5b) above.

a = (−1)a, for any number a. (Lesson 6.) Therefore −24 = (−1)24. And therefore according to the order of operations (Lesson 1), this is (−1)16 = −16.

c)   (−2)4 = +16, according to the Rule of Signs (Lesson 4).
The parentheses indicate that the base is −2.  See Problem 5c).

Example 1.  Negative base.

(−2)3 = (−2)(−2)(−2) = −8,

again according to the Rule of Signs.  Whereas,

(−2)4 = +16.

When the base is negative, and the exponent is odd, then the product is negative.  But when the base is negative, and the exponent is even, then the product is positive.

Problem 7.   Evaluate.

 a) (−1)2 = 1 b) (−1)3 = −1 c) (−1)4 = 1 d) (−1)5 = −1 e) (−1)100 = 1 f) (−1)253 = −1 g) (−2)4 = 16 h) (−2)5 = −32

Problem 8.   Rewrite using exponents.

 a) xxxxxx = x6 b) xxyyyy = x2y4 c) xyxxyx = x4y2 d) xyxyxy = x3y3

Problem 9.   Rewrite using exponents.

 a) (x + 1)(x + 1) = (x + 1)2 b) (x − 1)(x − 1)(x − 1) = (x − 1)3 c) (x + 1)(x − 1)(x + 1)(x − 1) = (x + 1)2(x − 1)2 d) (x + y)(x + y)2 = (x + y)3

Three rules

Rule 1.  Same Base

aman = am + n

"To multiply powers of the same base, add the exponents."

For example,  a2a3 = a5.

Why do we add the exponents?  Because of what the symbols mean.   Problem 4a.

Example 2.   Multiply  3x2· 4x5· 2x

Solution.   The problem means (Lesson 5):  Multiply the numbers, then combine the powers of x :

3x2· 4x5· 2x = 24x8

Two factors of x -- x2 -- times five factors of x -- x5 -- times one factor of x, produce a total of 2 + 5 + 1 = 8 factors of x :  x8.

Problem 10.   Multiply.  Apply the rule Same Base.

 a) 5x2· 6x4  = 30x6 b) 7x3· 8x6 = 56x9 c) x· 5x4 = 5x5 d) 2x· 3x· 4x = 24x3 e) x3· 3x2· 5x = 15x6 f) x5· 6x8y2 = 6x13y2 g) 4x· y· 5x2· y3 = 20x3y4 h) 2xy· 9x3y5 = 18x4y6 i) a2b3a3b4 = a5b7 j) a2bc3b2ac = a3b3c4 k) xmynxpyq = xm + pyn+ q l) apbqab = ap + 1bq + 1

Problem 11.   Distinguish the following:

x· x   and   x + x.

x· x = x².   x + x = 2x.

Example 3.   Compare the following:

a)  x· x5             b)  2· 25

Solution.

a)   x· x5 = x6

b)   2· 25 = 26

Part b) has the same form as part a).  It is part a) with x = 2.

One factor of 2 multiplies five factors of 2  producing six factors of 2.

2· 2 = 4 is not correct here.

Problem 12.   Apply the rule Same Base.

 a) xx7 = x8 b) 3· 37 = 38 c) 2· 24· 25 = 210 d) 10· 105 = 106 e) 3x· 36x6 = 37x7

Problem 13.   Apply the rule Same Base.

 a) xnx2 = xn + 2 b) xnx = xn + 1 c) xnxn = x2n d) xnx1 − n = x e) x· 2xn − 1 = 2xn f) xnxm = xn + m g) x2nx2 − n = xn + 2

Rule 2:  Power of a Product of Factors

(ab)n = anbn

"Raise each factor to that same power."

For example,  (ab)3 = a3b3.

Why may we do that?  Again, according to what the symbols mean:

(ab)3 = ab· ab· ab = aaabbb = a3b3.

The order of the factors does not matter:

ab· ab· ab = aaabbb.

Problem 14.   Apply the rules of exponents.

 a) (xy)4 = x4y4 b) (pqr)5 = p5q5r5 c) (2abc)3 = 23a3b3c3
 d)   x3y2z4(xyz)5 = x3y2z4· x5y5z5   Rule 2, = x8y7z9   Rule 1.

Rule 3:   Power of a Power

(am)n = amn

"To take a power of a power, multiply the exponents."

For example,  (a2)3 = a2· 3 = a6.

Why do we do that?  Again, because of what the symbols mean:

(a2)3 = a2a2a2 = a3· 2 = a6

Problem 15.   Apply the rules of exponents.

 a) (x2)5 = x10 b) (a4)8 = a32 c) (107)9 = 1063

Example 4.   Apply the rules of exponents:   (2x3y4)5

Solution.   Within the parentheses there are three factors:  2,  x3, and y4.   According to Rule 2 we must take the fifth power of each one.  But to take a power of a power, we multiply the exponents.  Therefore,

(2x3y4)5 = 25x15y20

Problem 16.   Apply the rules of exponents.

 a) (10a3)4 = 10,000a12 b) (3x6)2 = 9x12 c) (2a2b3)5 = 32a10b15 d) (xy3z5)2 = x2y6z10 e) (5x2y4)3 = 125x6y12

f)    (2a4bc8)6  = 64a24b6c48

Problem 17.   Apply the rules of exponents.

 a) 2x5y4(2x3y6)5  = 2x5y4· 25x15y30 = 26x20y34

b)  abc9(a2b3c4)8  = abc9· a16b24c32 = a17b25c41

Problem 18.   Use the rules of exponents to calculate the following.

 a) (2· 10)4 = 24· 104 = 16· 10,000 = 160,000

b)   (4· 102)3 = 43· 106 = 64,000,000

c)   (9· 104)2 = 81· 108 = 8,100,000,000

Example 5.   Square x4.

Solution.   (x4)2 = x8.

Thus to square a power, double the exponent.

Problem 19.   Square the following.

 a) x5 = x10 b) 8a3b6 = 64a6b12 c) −6x7 = 36x14 d) xn = x2n

Part c) illstrates:  The square of a number is always positive.

(−6)(−6) = +36.   The Rule of Signs.

Except 02 = 0.

Problem 20.   Apply a rule of exponents -- if possible.

 a) x2x5 = x7,  Rule 1. b) (x2)5 = x10,  Rule 3.
 c)   x2 + x5 Not possible.  The rules of exponents apply onlyto multiplication.

In summary:  Add the exponents when the same base appears twice:  x2x4 = x6.  Multiply the exponents when the base appears once -- and in parentheses:  (x2)5 = x10.

Problem 21.   Apply the rules of exponents.

 a) (xn)n = xn· n = xn² b) (xn)2 = x2n

Problem 22.   Apply a rule of exponents or add like terms -- if possible.

a)   2x2 + 3x4   Not possible. These are not like terms (Lesson 1).

b)   2x2· 3x4 = 6x6.  Rule 1.

c)   2x3 + 3x3  = 5x3.   Like terms.  The exponent does not change.

d)   x2 + y2   Not possible.  These are not like terms.

e)   x2 + x2  = 2x2.  Like terms.

f)   x2· x2  = x4.  Rule 1.

g)   x2· y3  Not possible.  Different bases.

h)   2· 26  = 27.  Rule 1.

i)   35 + 35 + 35 = 3· 35 (On adding those like terms) = 36.

We will continue the rules of exponents in Lesson 21.

Next Lesson:  Multiplying out. The distributive rule.

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