algebraic expressions – rules for exponents let’s review the rules for exponents you learned in...
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Algebraic Expressions – Rules for Exponents
Let’s review the rules for exponents you learned in Algebra 1 :
n
n
nmnm
nmn
m
nmnm
aa
aa
aa
a
aaa
1
Algebraic Expressions – Rules for Exponents
Let’s review the rules for exponents you learned in Algebra 1 :
n
n
nmnm
nmn
m
nmnm
aa
aa
aa
a
aaa
1
- When you multiply like variables you ADD their exponents
Algebraic Expressions – Rules for Exponents
Let’s review the rules for exponents you learned in Algebra 1 :
n
n
nmnm
nmn
m
nmnm
aa
aa
aa
a
aaa
1
- When you multiply like variables you ADD their exponents
- When you divide like variables you SUBTRACT their exponents
Algebraic Expressions – Rules for Exponents
Let’s review the rules for exponents you learned in Algebra 1 :
n
n
nmnm
nmn
m
nmnm
aa
aa
aa
a
aaa
1
- When you multiply like variables you ADD their exponents
- When an exponent is raised to another exponent, you MULTIPLY exponents
- When you divide like variables you SUBTRACT their exponents
Algebraic Expressions – Rules for Exponents
Let’s review the rules for exponents you learned in Algebra 1 :
n
n
nmnm
nmn
m
nmnm
aa
aa
aa
a
aaa
1
- When you multiply like variables you ADD their exponents
- When an exponent is raised to another exponent, you MULTIPLY exponents
- When you divide like variables you SUBTRACT their exponents
- When a variable has a negative exponent, you change its position in a fraction and the exponent becomes positive.
Algebraic Expressions – Rules for Exponents
Let’s review the rules for exponents you learned in Algebra 1 :
n
n
nmnm
nmn
m
nmnm
aa
aa
aa
a
aaa
1
- When an exponent is raised to another exponent, you MULTIPLY exponents
- When you divide like variables you SUBTRACT their exponents
- When a variable has a negative exponent, you change its position in a fraction and the exponent becomes positive.
75252 xxxx
Algebraic Expressions – Rules for Exponents
Let’s review the rules for exponents you learned in Algebra 1 :
n
n
nmnm
nmn
m
nmnm
aa
aa
aa
a
aaa
1
- When an exponent is raised to another exponent, you MULTIPLY exponents
- When a variable has a negative exponent, you change its position in a fraction and the exponent becomes positive.
75252 xxxx
3585
8
xxx
x
Algebraic Expressions – Rules for Exponents
Let’s review the rules for exponents you learned in Algebra 1 :
n
n
nmnm
nmn
m
nmnm
aa
aa
aa
a
aaa
1
- When a variable has a negative exponent, you change its position in a fraction and the exponent becomes positive.
75252 xxxx
3585
8
xxx
x
124343 xxx
Algebraic Expressions – Rules for Exponents
Let’s review the rules for exponents you learned in Algebra 1 :
n
n
nmnm
nmn
m
nmnm
aa
aa
aa
a
aaa
1
75252 xxxx
3585
8
xxx
x
124343 xxx
33 1
xx
Algebraic Expressions – Rules for Exponents
Let’s review the rules for exponents you learned in Algebra 1 :
n
n
nmnm
nmn
m
nmnm
aa
aa
aa
a
aaa
1
75252 xxxx
3585
8
xxx
x
124343 xxx
33 1
xx
What we are going to do in Pre – Calc is take these rules and combine them into multi – step problems that might have all four rules utilized.
Algebraic Expressions – Rules for Exponents
Example # 1 : cbbaa 22342
Algebraic Expressions – Rules for Exponents
Example # 1 : cbbaa 22342
Since everything is multiplied, we can multiply all integers, and all like variables.
cba
cbbaa
2231
223
8
42
Algebraic Expressions – Rules for Exponents
Example # 1 : cbbaa 22342
cba
cba
cbbaa
44
2231
223
8
8
42
Algebraic Expressions – Rules for Exponents
Example # 1 : cbbaa 22342
cba
cba
cbbaa
44
2231
223
8
8
42
Example # 2 :935236 xxxx nnmm
Algebraic Expressions – Rules for Exponents
Example # 1 : cbbaa 22342
cba
cba
cbbaa
44
2231
223
8
8
42
Example # 2 :935236 xxxx nnmm
Don’t PANIC !!! The rule for multiplying like variables still applies. Exponents can be algebraic, integers, or fractions
Algebraic Expressions – Rules for Exponents
Example # 1 : cbbaa 22342
cba
cba
cbbaa
44
2231
223
8
8
42
Example # 2 :935236 xxxx nnmm
Don’t PANIC !!! The rule for multiplying like variables still applies. Exponents can be algebraic, integers, or fractions
935236 xxxx nm We will still ADD exponents
Algebraic Expressions – Rules for Exponents
Example # 1 : cbbaa 22342
cba
cba
cbbaa
45
2241
223
8
8
42
Example # 2 :935236 xxxx nnmm
Now just treat the exponent like an algebraic expression where we combine like terms…
462
935236
xx
xxxx
nm
nm We will still ADD exponents
Algebraic Expressions – Rules for Exponents
Example # 3 : 3 23224 yx
When you have imbedded parentheses and exponents outside, start with the innermost set and work your way “out”.
Algebraic Expressions – Rules for Exponents
Example # 3 : 3 23224 yx
Evaluate this first, applying the exponent outside to ALL of the terms inside.
3 2322224 yx
Algebraic Expressions – Rules for Exponents
Example # 3 : 3 23224 yx
3 64
3 23222
44
24
yx
yx
Algebraic Expressions – Rules for Exponents
Example # 3 : 3 23224 yx
I like to multiply what is inside the brackets before I apply the outside exponent.
3 64
3 64
3 23222
16
44
24
yx
yx
yx
Algebraic Expressions – Rules for Exponents
Example # 3 : 3 23224 yx
Apply the outside exponent.
36343
3 64
3 64
3 23222
16
16
44
24
yx
yx
yx
yx
Algebraic Expressions – Rules for Exponents
Example # 3 : 3 23224 yx
181218123
36343
3 64
3 64
3 23222
4096or 16
16
16
44
24
yxyx
yx
yx
yx
yx
Algebraic Expressions – Rules for Exponents
Example # 4 :5
23
6
3
ab
ba There are two ways to attack this problem.
1. Apply the division rule using negative exponents.
2. Use the negative exponent rule first, then apply the division rule.
Algebraic Expressions – Rules for Exponents
Example # 4 :5
23
6
3
ab
ba There are two ways to attack this problem.
1. Apply the division rule using negative exponents.
2. Use the negative exponent rule first, then apply the division rule.
32
5213
2
16
3
ba
ba
Algebraic Expressions – Rules for Exponents
Example # 4 :5
23
6
3
ab
ba There are two ways to attack this problem.
1. Apply the division rule using negative exponents.
2. Use the negative exponent rule first, then apply the division rule.
2
53
6
3
ab
ba
I applied the negative exponent rule and moved any variable with a negative exponent to the other part of the fraction and made the exponent positive…
Algebraic Expressions – Rules for Exponents
Example # 4 :5
23
6
3
ab
ba There are two ways to attack this problem.
1. Apply the division rule using negative exponents.
2. Use the negative exponent rule first, then apply the division rule.
32
2513
2
53
2
16
36
3
ba
ba
ab
ba
Apply the normal division of like variables rule…
Algebraic Expressions – Rules for Exponents
Example # 5 : 3
1
21
21
43
236
yx
yx This problem has a few rules to apply, apply the exponent to each parentheses first…
Algebraic Expressions – Rules for Exponents
Example # 5 : 3
1
21
21
43
236
yx
yx This problem has a few rules to apply, apply the exponent to each parentheses first…
34
41
31
31
21
21
21
21
63643
2
xy
xy
yx
yx
When you have an integer to the ½ power, it is the square root of the integer.
Algebraic Expressions – Rules for Exponents
Example # 5 : 3
1
21
21
43
236
yx
yx This problem has a few rules to apply, apply the exponent to each parentheses first…
34
41
34
41
31
31
21
21
21
21
11
43
2
6
636
yx
xy
xy
yx
yx
Now you divide your variables…
Algebraic Expressions – Rules for Exponents
Example # 5 : 3
1
21
21
43
236
yx
yx This problem has a few rules to apply, apply the exponent to each parentheses first…
1213
1213
34
41
34
41
31
31
21
21
21
21
66
6
636
11
43
2
yy
yx
xy
xy
yx
yx
Anything to the zero power = 1, and it is not necessary to change improper fractions to mixed numbers…
Algebraic Expressions – Rules for Exponents
Example # 6 :1
213
532
4
12
zyx
zyx
Algebraic Expressions – Rules for Exponents
Example # 6 :1
213
532
4
12
zyx
zyx On this one I would reduce any integer fraction first…
Algebraic Expressions – Rules for Exponents
Example # 6 :
1
213
532
1
213
532
3
4
12
zyx
zyx
zyx
zyx On this one I would reduce any integer fraction first…
Algebraic Expressions – Rules for Exponents
Example # 6 :
213
5321
121113
1513121
1
213
532
1
213
532
3
3
3
4
12
zyx
zyx
zyx
zyx
zyx
zyx
zyx
zyx
Now apply the negative exponent outside…
Algebraic Expressions – Rules for Exponents
Example # 6 :
231
532
213
5321
121113
1513121
1
213
532
1
213
532
3
3
3
3
4
12
zyy
zxx
zyx
zyx
zyx
zyx
zyx
zyx
zyx
zyx
Move your negative exponent variables…
Algebraic Expressions – Rules for Exponents
Example # 6 :
231
532
213
5321
121113
1513121
1
213
532
1
213
532
3
3
3
3
4
12
zyy
zxx
zyx
zyx
zyx
zyx
zyx
zyx
zyx
zyx
Apply multiplication and division of variables rules…
4
35
31
2532
33 y
zx
y
zx