MATH 31 LESSONS PreCalculus 2. Powers. A. Power Laws Terminology: b x.

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


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


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


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


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


33 7 94 xx

31

37

94 xx

31

31

31

37

31

94 xxx

94 231

xx

Don’t stop here.


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


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.


543

2451

xxx

543 245 xxx

5554535 245 xxxx

24525 xxx

381

5 xx

x


8x

x


24

8x

x

248 xx

Convert to exponential notation

Bring all powers up to the top

24

8x

x

248 xx

42444 8 xxx


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.


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


44 9

3

2

2

3xx

41

49

3

2

2

3xx


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.


4

14

14

14

94

149

6

1xxx

496

1 241

xx

23236

1 4 xxx


6

3

40 x

x


5

6

3

40 x

x

xx5

6

3

40 1


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.


11111 910015

2 xxx

21 910015

2xx

xxx

31031015

2


235

2 595 xx


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.


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


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.


Then,

ABBA 34 53

73121274 53 xxxx

Don’t forget to use brackets


Then,

ABBA 34 53

73121274 53 xxxx

213121274 53 xxxx

Simplify inside the bracket


Then,

ABBA 34 53

73121274 53 xxxx

213121274 53 xxxx

2051274 53 xxx

412720 53 xxx

MATH 31 LESSONS PreCalculus 2. Powers. A. Power Laws Terminology: b x.

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Transcript of MATH 31 LESSONS PreCalculus 2. Powers. A. Power Laws Terminology: b x.