Page 31 - Calculus Workbook For Dummies

15
Chapter 1: Getting Down the Basics: Algebra and Geometry

Solutions for This Easy Elementary Stuff

5 5 0
a Solve = ?. is undefined! Don’t mix this up with something like , which equals zero. Note
0 0 8
rise 5
that if you think about these two fractions as examples of slope ( run ), 0 has a rise of 5 and a
run of 0 which gives you a vertical line that has sort of an infinite steepness or slope (that’s why
it’s undefined). Or just remember that it’s impossible to drive up a vertical road and so it’s
0
impossible to come up with a slope for a vertical line. The fraction , on the other hand, has a
8
rise of 0 and a run of 8, which gives you a horizontal line that has no steepness at all and thus
has the perfectly ordinary slope of zero. Of course, it’s also perfectly ordinary to drive on a
horizontal road.

0
b = 0 (See solution to problem 1.)
10
3 a + b a + b
c Does equal ? No. You can’t cancel the 3s.
3 a + c a + c
You can’t cancel in a fraction unless there’s an unbroken chain of multiplication running across
the entire numerator and ditto for the denominator.

3 a + b b
d Does equal ? No. You can’t cancel the 3as. (See previous Warning.)
3 a + c c
4 ab ab
e Does equal ? Yes. You can cancel the 4s because the entire numerator and the entire
4 ac ac
denominator are connected with multiplication.
4 ab b
f Does equal ? Yes. You can cancel the 4as.
c
4 ac
1
g Rewrite x - 3 without a negative power.
x 3
4
h Does abch equal a b c ? Yes. Exponents do distribute over multiplication.
4
4
4
^
4 4 4 4
b
i Does a + + ch equal a + b + c ? No! Exponents do not distribute over addition
^
(or subtraction).
When you’re working a problem and can’t remember the algebra rule, try the problem with
numbers instead of variables. Just replace the variables with simple, round numbers and work
out the numerical problem. (Don’t use 0, 1, or 2 because they have special properties that can
mess up your example.) Whatever works for the numbers will work with variables, and what-
ever doesn’t work with numbers won’t work with variables. Watch what happens if you try this
problem with numbers:
4 ? 4 4 4
4
6 =
^ 3 + + h 3 + 4 + 6
?
4
13 = 81 + 256 + 1296
28 ,561 ! 1633
j Rewrite 3 4 x with a single radical sign. 3 4 x = 12 x
2
2
k Does a + b equal a + b? No! The explanation is basically the same as for problem 9. Consider
/ 1 2
2
2
2
2
this: If you turn the root into a power, you get a + b = _ a + b i . But because you can’t
/ 1 2 / 1 2 / 1 2
2
2
2
2
2
2
distribute the power, a + b i ! _ a i + _ b i , or a + , b and thus a + b ! a + b.
_
c
l Rewrite log b c= as an exponential equation. a = b
a
a
m Rewrite log a - log b with a single log. log c
c
b
c