Elementary Abstract Algebra- Examples and Applications, 2019a

1.3 TEST YOURSELF 23
Exercise 1.2.19. For each of the above items (I–V), give a general equation
(using variables) that expresses the rule. For example one possible answer
to (II) is: x y+z = x y · x z .
♦
Exercise 1.2.20. Write an equation that shows another way to express a
number raised to a power that is the difference of two numbers. ♦
1.3 Test yourself
Test yourself with the following exercises. If you feel totally lost, I strongly
recommend that you improve your basic algebra skills before continuing with
this course. Trying to do higher math without a confident mastery of basic
algebra is like trying to play baseball without knowing how to throw and
catch.
Exercise 1.3.1. Simplify the following expressions. Factor whenever possible
(a) 2 4 4 2
(b) 39
(c)
9 3
( 5
9
) 7 ( 9
5
(d) a5
a 7 · a3
a
(e) x(y − 1) − y(x − 1)
) 6
♦
Exercise 1.3.2. Same instructions as the previous exercise. These examples
are harder. (Hint: It’s usually best to make the base of an exponent as
simple as possible. Notice for instance that 4 7 =(2 2 ) 7 =2 14 .)
(a) 6 1/2·21/6 · 3 3/2 · 2 1/3
(b) (9 3 )(4 7 ) ( )
1 8 ( 1
) 6
2 12
(c) 4 5 · 2 3 · ( 1
2) 5 ·
( 1
4
) 3
(d) 2 3 · 3 4 · 4 5 · 2 −5 · 3 −4 · 4 −3
(e)
x(x − 3) + 3(3 − x)
(x − 3) 2 ♦