Download EQ: How do I operate with exponents and special cases? Bellwork

August 30, 2012
EQ: How do I operate with exponents and special cases?
Bellwork Quiz: Simplify each expression and leave in exponential
form.
1. 58 x 512
2. 99
94
3. (56)3
8 x 42
4
4.
(42)3
Simplify each expression and leave in standard form.
5. 1270
6. (-4)
4
August 30, 2012
EQ: How do I operate with exponents and special cases?
Bellwork Quiz: Identify each relationship below as a relation, a
function, or both
1. (1, 2), (2, 3), (3, 4), (4, 5), (5, 6)
2.
x
y
6
5
7
5
8
10
5
5
3. (5, 6), (5, 7), (5, 8), (5, 9), (5, 10)
4. 1
2
3
4
5
6
4
5
6
7
8
9
5.
August 30, 2012
So, we've learned the rules for operating with exponents, but sometimes
there are special cases that we have to consider. these aren't so much
unique cases as they are cases in which multiple rules must be applied
in the correct order. Before we look at those cases, though, let's make
sure we remember the basic rules:
Rule # 1: ax x ay = a(x+y)
x
Rule # 2: a = a(x-y)
ay
Rule # 3: (ax)y = a(x*y)
Rule #4: a0 = 1
Rule #5: a-x =
1
ax
August 30, 2012
Special Case #1: What if our exponential expressions don't have
the same base?
Consider the following expression:
93 x 34
Based on our rules, there is nothing we can do here, right? Well, before
we give up, we need to make sure that we can't re-write our expression
in a way that allows us to follow the rules. Is there a way to write 9 as a
power of 3?
2
9=3
In mathematics we are allowed to substitute one expression for another
as long as those expressions are equivalent. In this case, 9 is equivalent
to 32, so we can use them interchangeably. When we substitute into our
original expression, we get:
23
4
(3 ) x 3
This is still a little complicated, but at least it follows the rules:
36 x 34
310
So, 93 x 34 = 310
By rule 3
By rule 1
Try checking out answer with a calculator!
August 30, 2012
Special Case #2: What if we have multiple exponential expression
and multiple rules to follow?
Ever since 5th grade, or maybe earlier, you've been hearing about
PEMDAS. Well, PEMDAS is one of the great things in mathematics that
works almost every time. If you have multiple operations to perform, all
you have to do is follow the prder of operations. Consider the expression
below:
6
(5 ) 3
(
(
(53)
In this case, we have multiple expressions and multiple operations. So
let's follow the order of operations:
P - Parentheses: We have three sets of parentheses. the smaller two
are really unnecessary. We could expand those expressions into
standard form, but that wouldn't be very helpful. The large set of
parentheses tells us to take everything inside those parentheses to the
3rd power. that gives us:
518
59
This is a little easier to deal with. using our division rule, we get:
59 as our answer!
August 30, 2012
Special Case #3: What if I don't have any numbers?
The higher you get in mathematics, the fewer actual numbers you are
going to deal with. More often, you will deal with variables instead.
Consider the expression below:
x5 * x4
x3 * x2
Variables follow the exact same rules as numbers do, so we can treat
this just like we did the last problem. All we do is follow our 5 basic rules
while keeping the order of operations in mind.
x5 * x 4
=
3
2
x *x
4
x9
= x
x5
August 30, 2012
Special Case #4: What if I have both exponential and nonexponential terms in my expression?
In algebra, we do something called collecting like terms. While not
exactly the same thing, we can do something similar when looking at
exponential expressions. We can work with only the pieces of the
expression that seem to fit together. Consider the expression below:
3b9
5
18b
At first glance, it's a little intimidating. What about if I wrote it like this?
3b9
5
18b
We know how to simplify a fraction, and that's all we have to do here:
1b9
6b5
Now we deal with the exponents and get:
4
Which we could also write as: 1 b
6
1b4
6
August 30, 2012
Special Case #5: What's the deal with negative exponents?
We have a pretty simple rule for dealing with negative exponents. the
problem is, it gets hard to figure out what to do with the negative sign
when you are already working with a fraction.
The first thing you need to know is that a truly simplified exponential
expression has only positive exponents. So, before an expression is
complete, we have to convert all of the negative exponents to fractions.
Look at this expression:
g6
1
= g-4 =
g10
g4
What about a harder one?
4k13
16k20
=
1k13
20
4k
=
1k
4
-7
=
1
4k7
August 30, 2012
Special Case #6: More negative exponents...
So what if we have a negative exponent in the denominator?
Consider the following expression:
If we simplify the fraction, we get:
5
25x-5
1
-5
5x
Now, we learned last year that we can think of a negative exponent as
an opposite sign. The same is true here. A negative exponent simply
means to move the exponent expression from the denominator to the
numerator, or to put it on the opposite side of the fraction. So our
expression becomes:
1x5
5
which equals
x5
5
which equals... 1 x5
5
August 30, 2012
Practice Problems: The Easy Sheet
4. (36)(32)2 = (36)(34) = 310
5. ((52)3)2 = (56)2 = 512
6. (35)(9) = (35)(32) = 37
(52)(58)
510
7.
=
= 57
3
53
5
8
28
28
8. 2
=
= 22 = 4
=
23
(2 )
43
26
9.
(
(1516)
(15
11
)
4
(
3 7 2
10 2
12
1. (2 )(2 )(2 ) = 2 (2 ) = 2
8
2. 12
= 124
124
3. 62 × 69 = 611
= (155)4
= 15
20
August 30, 2012
Practice Sheet 2: A little Harder
1. a4× a3 = a7
2. n5× n4 = n9
3. x2 × x9 = x11
4. y × y6 × y2 = y9
5.
y3
y2
7
a
6.
a3
7.
=y
= a4
1
a6
= a-2 =
a8
a2
a7
a3
August 30, 2012
Practice: Simplify each of the following exponential expressions. Your
final expression should have no negative exponents.
1. (3n4)2
n2
2. 15(x3)4
3(x5)3
3. (2a3b4c5)2
4(a2b2c2)4
4. (4p4r2s6)3
(2p2rs3)6