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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   Same Base.

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

The powers of 10 have as many 0's as the exponent of 10.

Example 4.   Square x4.

Solution.   (x4)2 = x8.

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 never negative.

(−6)(−6) = +36.   The Rule of Signs.

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.

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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