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13

EXPONENTS

Powers of a number

Section 2

Three rules of exponents

WHEN ALL THE FACTORS ARE EQUAL -- 2· 2· 2· 2 -- we call the product a power of that number (Lesson 1).

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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 -- the number being repeated multiplied -- 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)3a2· 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

Section 2:  Three rules of exponents


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