math 31 lessons precalculus 2. powers. a. power laws terminology: b x
Post on 12-Jan-2016
215 Views
Preview:
TRANSCRIPT
MATH 31 LESSONS
PreCalculus
2. Powers
A. Power Laws
Terminology:
bx
bx
base
bx exponent
bxthe base and exponent together form a power
Power Laws
1. xa xb = ?
xa xb = xa + b
If the bases are the same ...
Keep the base the same and add the exponents
xa xb = xa + b
Similarly,
xa ÷ xb = xa b
e.g. Simplify
4
75
y
yy
4
75
y
yy
4
75
y
y
4
12
y
y
4
75
y
yy
4
75
y
y
4
12
y
y
412 y 8y
(xa) b = xa b = xa b
Multiply the exponents
(xa) b = xa b = xa b
Similarly,
(x y)n = xn yn
n
nn
y
x
y
x
Note: This works only for multiplication and division.
It does NOT work for addition or subtraction.
(a + b2)3 ≠ a3 + b6
(x - y)5 ≠ x5 - y5
e.g. Simplify
2
3
5
c
ba
2
3
5
c
ba
23
225
c
ba
2
3
5
c
ba
23
225
c
ba6
210
c
ba
Power Laws
3. x0 = ? x1 = ?
x0 = 1 x1 =x
1
x0 = 1 x1 =
Similarly,
x
1
aa
xx
1 a
ax
x
1
If you move the power from the top to the bottom (or bottom to the top), it gets the opposite exponent
e.g. Simplify
Express your answer with only positive exponents.
03
2
8
5
zy
x
03
2
8
5
zy
x
18
53
2
y
x2
3
8
5
x
y
Power Laws
4. = ?bax
Remember, the root is always the one on the
bottom of the fraction.
abb arootpower
ba
xxxx
e.g. Evaluate
23
25
32
32525 The root is on the bottom
of the fraction
= 53 = 125
323
2525
Ex. 1 Simplify
Answer with positive exponents.
Try this example on your own first.Then, check out the solution.
zx
yx
yx
zyx3
52
41
27
3
10
8
6
zx
yx
yx
zyx3
52
81
27
3
10
8
6
zyx
zyx84
35
24
60
zx
yx
yx
zyx3
52
81
27
3
10
8
6
zyx
zyx84
35
24
60
059
2
5zyx
zx
yx
yx
zyx3
52
81
27
3
10
8
6
zyx
zyx84
35
24
60
059
2
5zyx
5
9
2
5
y
x
Ex. 2 Simplify
Try this example on your own first.Then, check out the solution.
23 468 yx
23 468 yx
32
468 yx
23 468 yx
32
468 yx
3243
2632
8
yx
23 468 yx
32
468 yx
3243
2632
8
yx
38
3122
3 8
yx
23 468 yx
32
468 yx
3243
2632
8
yx
38
3122
3 8
yx
38
44
yx
B. Factoring Power Expressions
Method:
Convert all variables to exponential notation
- bring all powers to the numerator
Convert all fractions to LCD
Factor out the smallest power
- remove the factor by dividing
(subtracting the exponents)
- this should leave the exponents positive
Ex. 3 Factor completely xxx 35 2
Try this example on your own first.Then, check out the solution.
xxx 35 2
21
23
25
2 xxx Convert to exponential notation
xxx 35 2
21
23
25
2 xxx
2
12
12
12
32
12
52
12 xxxx
Factor out the smallest power
When you divide, you subtract exponents
xxx 35 2
21
23
25
2 xxx
2
12
12
12
32
12
52
12 xxxx
12221
xxx
Don’t stop here.
What else can you do?
xxx 35 2
21
23
25
2 xxx
2
12
12
12
32
12
52
12 xxxx
12221
xxx
1121
xxx 221
1 xx
Ex. 4 Factor completely 33 7 94 xx
Try this example on your own first.Then, check out the solution.
33 7 94 xx
31
37
94 xx Convert to exponential notation
33 7 94 xx
31
37
94 xx
31
31
31
37
31
94 xxx
Factor out the smallest power
33 7 94 xx
31
37
94 xx
31
31
31
37
31
94 xxx
94 231
xx
Don’t stop here.
What else can you do?
33 7 94 xx
31
37
94 xx
31
31
31
37
31
94 xxx
94 231
xx
32323 xxx
Ex. 5 Factor completely 543
2451
xxx
Try this example on your own first.Then, check out the solution.
543
2451
xxx
543 245 xxxConvert to exponential notation
Bring all powers up to the top
543
2451
xxx
543 245 xxx
5554535 245 xxxx
Factor out the smallest power.
Notice that to subtract 5 from the exponents, you add +5
543
2451
xxx
543 245 xxx
5554535 245 xxxx
24525 xxx
Don’t stop here.
What else can you do?
543
2451
xxx
543 245 xxx
5554535 245 xxxx
24525 xxx
381
5 xx
x
Ex. 6 Factor completely 24
8x
x
Try this example on your own first.Then, check out the solution.
24
8x
x
248 xx
Convert to exponential notation
Bring all powers up to the top
24
8x
x
248 xx
42444 8 xxx
Factor out the smallest power
Notice that to subtract 4 from the exponents, you add +4
24
8x
x
248 xx
42444 8 xxx
64 8 xx
Don’t stop here.
What else can you do?
64 8 xx
3234 2 xx
This is a difference of cubes
64 8 xx
3234 2 xx
222224 222 xxxx
A3 - B3 = (A - B) (A2 + AB + B2)
64 8 xx
3234 2 xx
222224 222 xxxx
4224
2421
xxxx
Ex. 7 Factor completely 44 9
3
2
2
3xx
Try this example on your own first.Then, check out the solution.
44 9
3
2
2
3xx
41
49
3
2
2
3xx
Convert to exponential notation
44 9
3
2
2
3xx
41
49
3
2
2
3xx
41
49
6
4
6
9xx Convert the coefficients
to the LCD
44 9
3
2
2
3xx
41
49
3
2
2
3xx
41
49
6
4
6
9xx
4
14
14
14
94
149
6
1xxx
Factor out the common coefficients and the lowest power
4
14
14
14
94
149
6
1xxx
496
1 241
xx
Don’t stop here.
What else can you do?
4
14
14
14
94
149
6
1xxx
496
1 241
xx
23236
1 4 xxx
Ex. 8 Factor completely 5
6
3
40 x
x
Try this example on your own first.Then, check out the solution.
5
6
3
40 x
x
xx5
6
3
40 1
Convert to exponential notation
Bring all powers to the top
5
6
3
40 x
x
xx5
6
3
40 1
xx15
18
15
200 1
Convert coefficients to LCD
5
6
3
40 x
x
xx5
6
3
40 1
xx15
18
15
200 1
11111 910015
2 xxx
Factor out the common coefficients and the lowest power
11111 910015
2 xxx
21 910015
2xx
Don’t stop here.
What else can you do?
11111 910015
2 xxx
21 910015
2xx
xxx
31031015
2
Ex. 9 Factor completely 32
235
2 595 xx
Try this example on your own first.Then, check out the solution.
Let A = x2 + 5
Then,
32
235
2 595 xx
Use substitution to remove the common binomial from the expression. Makes it simpler.
Let A = x2 + 5
Then,
32
235
2 595 xx
32
35
9AA
Let A = x2 + 5
Then,
32
235
2 595 xx
32
35
9AA
32
32
32
35
32
9AAA
Factor out the lowest power
Let A = x2 + 5
Then,
32
235
2 595 xx
32
35
9AA
32
32
32
35
32
9AAA
932
AA
Since A = x2 + 5
Then,
932
AA
Now, back substitute to return the expression to its original variable.
Since A = x2 + 5
Then,
932
AA
955 232
2 xx
Don’t forget to use brackets
Since A = x2 + 5
Then,
932
AA
955 232
2 xx
45 232
2 xx
Don’t stop here.
What else can you do?
Since A = x2 + 5
Then,
932
AA
955 232
2 xx
45 232
2 xx
225 32
2 xxx
Ex. 10 Factor completely
5463 127121274 xxxx
Try this example on your own first.Then, check out the solution.
Let A = x + 7 and B = 2x - 1
Then,
5463 127121274 xxxx
Use substitution to remove the common binomials from the expression. Makes it simpler.
Let A = x + 7 and B = 2x - 1
Then,
5463 127121274 xxxx
5463 124 BABA
Let A = x + 7 and B = 2x - 1
Then,
5463 127121274 xxxx
5463 124 BABA
ABBA 34 53
Since A = x + 7 and B = 2x - 1
Then,
ABBA 34 53
Now, back substitute to return the expression to its original variables.
Since A = x + 7 and B = 2x - 1
Then,
ABBA 34 53
73121274 53 xxxx
Don’t forget to use brackets
Since A = x + 7 and B = 2x - 1
Then,
ABBA 34 53
73121274 53 xxxx
213121274 53 xxxx
Simplify inside the bracket
Since A = x + 7 and B = 2x - 1
Then,
ABBA 34 53
73121274 53 xxxx
213121274 53 xxxx
2051274 53 xxx
412720 53 xxx
top related