ARITHMETICS OF EXPONENTS

Objectives:

• To recall how to multiply and divide powers of numbers or symbols such as x^n.
• To recall how to simplify expressions that involve powers.

Recall 1Rules for exponents.
These rules tell us how to multiply and divide powers of a number.

3^2 means 3*3,
3^5 means 3*3*3*3*3,
7^4 means 7*7*7*7,

and, more generally,

x^5 means x*x*x*x*x.

When we write 7^4, we call 4 the exponent and 7 the base.
The exponent is the number that tells you how many 7's to multiply together to make 7^4.
In general, the number x^n has an exponent n and base x.  This means that we must multiply x by itself n times in order to have the same number that x^n represents.  Clearly, this only makes sense if n is a positive number.

You may have forgotten the basic rules for multiplying exponents and raising exponents to powers.  Here are some general rules.

1.  x^m*x^n = x^(m+n),

2.  (x^m)^n = x^(m*n).

It is easy to convince yourself that the two rules above are true.  Take a few sample numbers -- let x = 3, m = 2 and n = 5.  The question is whether or not the equalities

3^2*3^5 = 3^(2+5)

and
(3^2)^5 = 3^(2*5)

hold.  The validity of these equations does not prove the general equations 1 and 2, but it does make them more believable.

Notice that the first equality may be written as

(3*3)*(3*3*3*3*3) = 3^7,
or as
3*3*3*3*3*3*3 = 3^7.

This last equality is true, by definition of 3^7.

What about the equality (3^2)^5 = 3^(2*5) ?  Convince yourself that it is true.

Recall 2: Zero and negative exponents.

3.  x^0 = 1
and that

4.

We shall not go into the reasons for this.  Just think of x^(-n) as short hand notation for 1/(x^n) . and x^(m-n) as short hand for  (x^m)/(x^n).

Recall 3: Products and quotients of powers.
What happens when you   take the product or quotient of a number raised to a power?

It is easy to see that the equalities below hold.

(3^4)*(5^4) = (3*5)^4,
and, more generally,

5.  (x^n)*(y^n) = (x*y)^n.

Also

and, more generally,

6.    .

Examples

Example 1.      Expand x^8.

Solution:          The symbol x^8 means multiply x by itself 8 times.  You get

x^8=x*x*x*x*x*x*x*x

Note

The symbol -x^8 generally means -(x^8), or -x*x*x*x*x*x*x*x.  However,
the symbol (-x)^8 means (-x)*(-x)*(-x)*(-x)*(-x)*(-x)*(-x)*(-x), which, in turn means x*x*x*x*x*x*x*x, because (-x)*(-x) = x*x.

Example 2.   Find other ways of writing (1/x)^5.

Solution:   The symbol  (1/x)^5. means multiply 1/x by itself 5 times.  You get
(1/x)^5= (1/x)*(1/x)*(1/x)*(1/x)*(1/x).

You may also use rule 6 to get  (1/x)^5=(1^5)/(x^5)  , which may be written as 1/(x^5).

You may now use rule 4 to get  1/(x^5)= x^(-5).

Example 3

(a) (x^7)*x = x^8.

(b) (2*x)^3*(2*x)^5 = (2*x)^(3+5) = (2*x)^8.

Note

1.  x^2 + x^3 cannot be written in any other way, since it is a sum of two powers, not a product.

2.  2^3*5^4 cannot be simplified, because the bases are different.

Example 4

EXERCISES

Simplify each of the following expressions:

1. 5*x^2*x^3

2. (5*x)^3*3*x^2

3  (9*x*y*z^3)^2

4 (5*x^3*3*x^2)*(3*x^4)

5 (5*x^3*3*y^2)*(3*x^4)

6 (3*a^3)^3

7 [(3*x)^5]/[(3*x)^(-5)]

8 -

10. Write

using only positive exponents.

11 Write

using only positive exponents.

12 Simplify

13 Write [(x^(-1)*y^(-1)]^(-1)  using only positive exponents.

14 Write [(x^(-1)+y^(-1)]^(-1)  using only positive exponents.

15 Simplify  .(x^2+x^0)*(1+x^2)^(-1).

16 Simplify
.