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Lesson 13, Section 2

Three Rules of Exponents

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Rule 1. Same base

Rule 2. Power of a product

Rule 3. Power of a power

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.   Section 1.

Example 1.   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 1.   Multiply.  Apply the rule Same Base.

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   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 2.   Distinguish the following:

x· x   and   x + x.

x· x = x².   x + x = 2x.

Example 2.   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 3.   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 4.   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 5.   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 6.   Apply the rules of exponents.

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

Example 3.   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 7.   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 8.   Apply the rules of exponents.

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

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

Problem 9.   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 4.   Square x4.

Solution.   (x4)2 = x8.

Thus to square a power, double the exponent.

Problem 10.   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 11.   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 only
to 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 12.   Apply the rules of exponents.

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

Problem 13.   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.

end

Next Lesson:  Multiplying out. The distributive rule.

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