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Exponentiation:

Exponentiation is a mathematical operation that involves an expression written as:

  an    

which indicates that a number (a) is multiplied by itself n times.
a is called the base and n is called the exponent. The expression is read as a to the power of n or a raised to the power of n. If the exponent is 2, it is usually read as square; if the exponent is 3, it is usually read as cubed.

To provide an example, the expression:

  42    

means that we will multiply 4 two times or

  4 x 4 = 16    

Another example would be 53

which means that we would multiply 5 three times or

  5 x 5 x 5 = 125    

The base that we may be asked to exponentiate may not necessarily be simply an integer. You may frequently encounter an expression, such as:

  (2 + 4)2    

where we will first add 2 + 4 which equals 6 and then square 6 to give us the result of 36. This is most often encountered when working with unknowns, but for discussion of unknowns, you will have to go to our Advanced Section.

What about exponentiation of a negative number? Well, if you have learned about multiplying of negative numbers, you know that multiplying two negative numbers results in a positive number. Consequently, if you are asked to square a negative number, such as:

  -52    

this will be the same as:

  -5 x -5    

and since we know that multiplying two negative numbers will yield a positive number, the answer will be +25.

What about cubing a negative number? Let us look at:

  -53    

That is the same as expressing it:

  -5 x -5 x -5    

If you multiply - 5 by - 5, as we have just seen, the result will be +25. You will then multiply:

  +25 x -5    

Now you are multiplying a positive number by a negative number, so the result will be negative and our answer will be:

  -125    

In exponentiation of negative numbers, if the exponent is an even number, the result will be positive. If the exponent is an odd number, the result will be negative.

What about if the exponent itself is negative?

  a-n    

will be the same as the expression

  1/an    

Rather than attempting to understand the reason for this, simply remember it. You will not encounter it too often anyway.

Exponents have traditionally been accepted only as integers. You may encounter an exponent that will be a fraction, however that is advanced math that will not be assigned until far higher grades. For the time being, get used to working with integers as exponents only!

Examples:

1)  
 43 =  
11)  
  (3 + 4.2)2 =
2)  
 72 =  
12)  
  (12.7 - 4.9)2 =
3)  
 63 =  
13)  
  (24.8 - 16.9)2 =
4)  
 33 =  
14)  
  (28.6 - 24.7)3 =
5)  
 54 =  
15)  
  (11.8 + 12.6)2 =
6)  
 23 =  
16)  
  (17.5 - 8.1)3 =
7)  
 4.22 =  
17)  
  1/42 =
8)  
 6.32 =  
18)  
  2/32 =
9)  
 9.82 =  
19)  
  (4/5 - 1/4)2 =
10)  
  3.63 =  
20)  
  (5/6 - 1/4)2 =


Answers:

1)  
  64  
11)  
  51.84
2)  
  49  
12)  
  60.84
3)  
  216  
13)  
  62.41
4)  
  27  
14)  
  59.319
5)  
  625  
15)  
  595.36
6)  
  8  
16)  
  830.584
7)  
  17.64  
17)  
  1/16
8)  
  39.69  
18)  
  4/9
9)  
  96.04  
19)  
  121/400
10)  
  46.656  
20)  
  49/144
 
 
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