1 rules for exponents

Algebraic Expressions – Rules for Exponents

Let’s review the rules for exponents you learned in Algebra 1 :

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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

44

2231

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