EXPONENTS AND RADICALS

 

Exponential notation is a simple way of writing a factor a number of times.  For example, 2 x 2 x 2 x 2 x 2 is "2 to the 5th power" and is written ,  where 2 is the base and 5 is the exponent.

Caution:  10  (This is a common error!)

 

 is also read as "five squared" and  is also read as "five cubed."

Note that the area of a square with edge of length 5 is 5x5 = , and the volume of a cube with edge of length 5 is 5x5x5=.

 

It might seem that  should equal zero since the exponent appears to indicate that 2 is not used as a factor.  However,

= 1, and for any base a, .

 

Caution: .  (This is a common error!)

 

 

RULES FOR EXPONENTS

 

Multiplying numbers with the same base:

Example:   x  = (2 x 2 x 2) x (2 x 2) =  =

Notice that we added the exponents.

Rule:

Example:

 

Finding a power of a power:

Example: = (2 x 2) x (2 x 2) x (2 x 2) =

Notice that we multiplied exponents.

Rule:

 

Division of numbers with the same base.

Example:

Note that we subtracted the exponents.

Rule:

 

Now we can use what we know about division of exponents to justify our statement that

 by using an example:

 

We'll show 1 = :

.

 

Exponent of a product:

 = (2 x 3) x (2 x 3) x (2 x 3) = (2 x 2 x 2) x (3 x 3 x 3) =

We can think of this as "distributing" an exponent among the factors of a product. 

Rule:

Caution:  (This is a common error!)

 

Note: (2x)3= (2x)(2x)(2x) = 8x3  and 2x3 = 2(x)(x)(x).

Caution:     (This is a common error!)

 

Can we simplify an expression with different bases and different exponents?

Example:  Simplify 

Answer:   can't be simplified.

Example:  Simplify 

 Answer: Note that 8 = .  Then =.

 

Example:  Find (-1)8

 

Note that (-1)2 is equal to 1.  In other words, every time we have -1 as a factor two times, we get +1.  since -1 is a factor an even number of times, (-1)8=+1.

 

Important rule:  A negative number is raised to an odd power will be negative, and a negative number raised to an even power will be positive.

 

 

Example:  Which is greater, (-3)11 or (-2)14?

With a calculator it may not be difficult to find the answer.  However, note that 11 is odd so that (-3)11 will be negative, and 14 is even so that (-2)14

will be positive.  Every positive number is greater than every negative number.  Therefore, (-2)14 > (-3)11.

 

Example:  If 0<x<1, which is greater x5 or x6?

Note that x6 is equal to (x5)x.  Therefore, no matter what the value of x5 is, the result of multiplying x5 by a positive fraction will be less than x5.

Important rule:  If x is a positive number less than 1 and if y is a positive integer, xy will be less than x.

 

Example:  Simplify (z2w4)3

Solution: (z2w4)3=(z2)3(w4)3=z6w12

 

 

MORE ABOUT EXPONENTS

 

Negative exponents

A number raised to a negative power is the reciprocal of the number raised to the absolute value of that power. 

That is, a-m= . 

 

Example:  Simplify

Answer:  x-4= .  Therefore,  =  = x4.

 

Example:  Simplify (v5)/(v-2)

 

Answer: (v5)/(v-2)= v5-(-2)= v7

 

 

Fractional exponents:  To find the value of a number raised to a fraction of the form 1/n, find the nth root of the number.

 

Example:  91/2= =3.

 

Example:  1251/3=5.  Recall that 53=125.

 

The rules that apply to exponents also apply to roots, since roots are just fractional exponents.

 

Example:  Simplify

 

Answer: ===

 

Example:  True of false: =

Answer:  Plug in numbers. ==5 and=4+3=7.  Note that 5 is not equal to 7. 

 

Caution:   (This is a common error!)

 

Example:  Simplify

 

Answer:  The perfect squares are 1, 4, 9, 16, 25, 36 ... .  In other words, they are the squares of the integers.  To simplify a problem like this, look for perfect squares that are factors of 75 and 48. We see that 25 is a factor of 75 and 16 is a factor or 48.

Then

=  

=

=

Example:

If 10 2x+2 = 1,000,000,000,000 then x = ?

 

A.  4

B.  5

C.  6

D.  7

E.  8

 

Since 1012 = 1,000,000,000,000 then 2x+2=12.  Solve for x:

2x+2=12

2x = 10

x = 5

 

Test taking trick:  Plug in answers.  In this case, where all answers are equally easy to plug in, it is not necessary to plug in all the answers.  In fact, only two answers need to be tried.  Start in the middle with (C) which is 6.  If this is too large, try (B), x= 5.  If this is too large, the answer must be A.  Should you plug it in to check?  NO!!  Go on to another problem.  Similarly, if (C) is too small, try (D) to determine if the answer is (D) or (E).

 

10 2(6)+2 = 10 14 = 100,000,000,000,000

 

This is too large so pick a smaller exponent.

 

10 2(5)+2 = 10 12 = 1,000,000,000,000.  This is the correct answer.