Exponents

In the notation

23

Exponents

In the notation

23this is the base

Exponents

In the notation

= 2 * 2 * 223this is the base

this is the exponent, or the power, which is the number of repetitions.

Exponents

In the notation

= 2 * 2 * 223this is the base

this is the exponent, or the power, which is the number of repetitions.

= 8

Exponents

In the notation

= 2 * 2 * 223this is the base

this is the exponent, or the power, which is the number of repetitions.

We say that “2 to the power 3 is 8” or that “2 to the 3rd power is 8.”

= 8

Exponents

In the notation

= 2 * 2 * 223this is the base

this is the exponent, or the power, which is the number of repetitions.

The Base-1 Rule: 1 any power = 1 * 1 * ..*1 = 1.

We say that “2 to the power 3 is 8” or that “2 to the 3rd power is 8.”

= 8

Exponents

In the notation

= 2 * 2 * 223this is the base

this is the exponent, or the power, which is the number of repetitions.

The Base-1 Rule: 1 any power = 1 * 1 * ..*1 = 1.The Blank-1Power : The expression x (with blank power) is x1, so 2 = 21, 7 = 71, etc.., i.e. we have one copy of x.

We say that “2 to the power 3 is 8” or that “2 to the 3rd power is 8.”

= 8

Exponents

In the notation

= 2 * 2 * 223this is the base

this is the exponent, or the power, which is the number of repetitions.

Example A. Calculate the following.

The Base-1 Rule: 1 any power = 1 * 1 * ..*1 = 1.The Blank-1Power : The expression x (with blank power) is x1, so 2 = 21, 7 = 71, etc.., i.e. we have one copy of x.

We say that “2 to the power 3 is 8” or that “2 to the 3rd power is 8.”

= 8

a. 3(4) b. 34 c. 43

Exponents

In the notation

= 2 * 2 * 223this is the base

this is the exponent, or the power, which is the number of repetitions.

Example A. Calculate the following.

The Base-1 Rule: 1 any power = 1 * 1 * ..*1 = 1.The Blank-1Power : The expression x (with blank power) is x1, so 2 = 21, 7 = 71, etc.., i.e. we have one copy of x.

We say that “2 to the power 3 is 8” or that “2 to the 3rd power is 8.”

= 8

= 12a. 3(4) b. 34 c. 43

Exponents

In the notation

= 2 * 2 * 223this is the base

this is the exponent, or the power, which is the number of repetitions.

Example A. Calculate the following.

The Base-1 Rule: 1 any power = 1 * 1 * ..*1 = 1.The Blank-1Power : The expression x (with blank power) is x1, so 2 = 21, 7 = 71, etc.., i.e. we have one copy of x.

We say that “2 to the power 3 is 8” or that “2 to the 3rd power is 8.”

= 8

= 12 = 3*3*3*3a. 3(4) b. 34 c. 43

Exponents

In the notation

= 2 * 2 * 223this is the base

this is the exponent, or the power, which is the number of repetitions.

Example A. Calculate the following.

The Base-1 Rule: 1 any power = 1 * 1 * ..*1 = 1.The Blank-1Power : The expression x (with blank power) is x1, so 2 = 21, 7 = 71, etc.., i.e. we have one copy of x.

We say that “2 to the power 3 is 8” or that “2 to the 3rd power is 8.”

= 8

= 12 = 3*3*3*3

= 9 9*

a. 3(4) b. 34 c. 43

Exponents

In the notation

= 2 * 2 * 223this is the base

this is the exponent, or the power, which is the number of repetitions.

Example A. Calculate the following.

The Base-1 Rule: 1 any power = 1 * 1 * ..*1 = 1.The Blank-1Power : The expression x (with blank power) is x1, so 2 = 21, 7 = 71, etc.., i.e. we have one copy of x.

We say that “2 to the power 3 is 8” or that “2 to the 3rd power is 8.”

= 8

= 12 = 3*3*3*3

= 9 9*= 81

a. 3(4) b. 34 c. 43

Exponents

In the notation

= 2 * 2 * 223this is the base

this is the exponent, or the power, which is the number of repetitions.

Example A. Calculate the following.

The Base-1 Rule: 1 any power = 1 * 1 * ..*1 = 1.The Blank-1Power : The expression x (with blank power) is x1, so 2 = 21, 7 = 71, etc.., i.e. we have one copy of x.

We say that “2 to the power 3 is 8” or that “2 to the 3rd power is 8.”

= 8

= 12 = 3*3*3*3

= 9 9*= 81

= 4 * 4 * 4a. 3(4) b. 34 c. 43

Exponents

In the notation

= 2 * 2 * 223this is the base

this is the exponent, or the power, which is the number of repetitions.

Example A. Calculate the following.

The Base-1 Rule: 1 any power = 1 * 1 * ..*1 = 1.The Blank-1Power : The expression x (with blank power) is x1, so 2 = 21, 7 = 71, etc.., i.e. we have one copy of x.

We say that “2 to the power 3 is 8” or that “2 to the 3rd power is 8.”

= 8

= 12 = 3*3*3*3

= 9 9*= 81

= 4 * 4 * 4a. 3(4) b. 34 c. 43

= 16 * 4= 64

Exponents

base

exponent

ExponentsWe write the quantity A multiplied to itself N times as AN, i.e.

A x A x A ….x A = AN

Example B.43

base

exponent

ExponentsWe write the quantity A multiplied to itself N times as AN, i.e.

A x A x A ….x A = AN

Example B.43 = (4)(4)(4) = 64

base

exponent

ExponentsWe write the quantity A multiplied to itself N times as AN, i.e.

A x A x A ….x A = AN

Example B.43 = (4)(4)(4) = 64 (xy)2

base

exponent

ExponentsWe write the quantity A multiplied to itself N times as AN, i.e.

A x A x A ….x A = AN

Example B.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy)

base

exponent

ExponentsWe write the quantity A multiplied to itself N times as AN, i.e.

A x A x A ….x A = AN

Example B.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2

base

exponent

ExponentsWe write the quantity A multiplied to itself N times as AN, i.e.

A x A x A ….x A = AN

Example B.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2

xy2

base

exponent

ExponentsWe write the quantity A multiplied to itself N times as AN, i.e.

A x A x A ….x A = AN

Example B.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2

xy2 = (x)(yy)

base

exponent

ExponentsWe write the quantity A multiplied to itself N times as AN, i.e.

A x A x A ….x A = AN

Example B.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2

xy2 = (x)(yy) –x2 = –(xx)

base

exponent

ExponentsWe write the quantity A multiplied to itself N times as AN, i.e.

A x A x A ….x A = AN

Example B.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2

xy2 = (x)(yy) –x2 = –(xx)

base

exponent

ExponentsWe write the quantity A multiplied to itself N times as AN, i.e.

A x A x A ….x A = AN

Example B.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2

xy2 = (x)(yy) –x2 = –(xx)

base

exponent

Exponents

Rules of Exponents

We write the quantity A multiplied to itself N times as AN, i.e.

A x A x A ….x A = AN

Example B.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2

xy2 = (x)(yy) –x2 = –(xx)

base

exponent

Exponents

Multiply-Add Rule: ANAK =AN+K Rules of Exponents

We write the quantity A multiplied to itself N times as AN, i.e.

A x A x A ….x A = AN

Example B.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2

xy2 = (x)(yy) –x2 = –(xx)

base

exponent

Exponents

Multiply-Add Rule: ANAK =AN+K Example C .

a. 5354

Rules of Exponents

We write the quantity A multiplied to itself N times as AN, i.e.

A x A x A ….x A = AN

Example B.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2

xy2 = (x)(yy) –x2 = –(xx)

base

exponent

Exponents

Multiply-Add Rule: ANAK =AN+K Example C .

a. 5354 = (5*5*5)(5*5*5*5)

Rules of Exponents

We write the quantity A multiplied to itself N times as AN, i.e.

A x A x A ….x A = AN

Example B.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2

xy2 = (x)(yy) –x2 = –(xx)

base

exponent

Exponents

Multiply-Add Rule: ANAK =AN+K Example C .

a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57

b. x5y7x4y6

Rules of Exponents

We write the quantity A multiplied to itself N times as AN, i.e.

A x A x A ….x A = AN

Example B.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2

xy2 = (x)(yy) –x2 = –(xx)

base

exponent

Exponents

Multiply-Add Rule: ANAK =AN+K Example C .

a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57

b. x5y7x4y6 = x5x4y7y6

Rules of Exponents

We write the quantity A multiplied to itself N times as AN, i.e.

A x A x A ….x A = AN

Example B.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2

xy2 = (x)(yy) –x2 = –(xx)

base

exponent

Exponents

Multiply-Add Rule: ANAK =AN+K Example C .

a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57

b. x5y7x4y6 = x5x4y7y6 = x9y13

Rules of Exponents

We write the quantity A multiplied to itself N times as AN, i.e.

A x A x A ….x A = AN

Example B.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2

xy2 = (x)(yy) –x2 = –(xx)

base

exponent

Exponents

Multiply-Add Rule: ANAK =AN+K Example C .

a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57

b. x5y7x4y6 = x5x4y7y6 = x9y13

Rules of Exponents

Divide-Subtract Rule: AN

AK = AN – K

We write the quantity A multiplied to itself N times as AN, i.e.

A x A x A ….x A = AN

Example D . 56

52 =

Example B.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2

xy2 = (x)(yy) –x2 = –(xx)

base

exponent

Exponents

Multiply-Add Rule: ANAK =AN+K Example C .

a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57

b. x5y7x4y6 = x5x4y7y6 = x9y13

Rules of Exponents

Divide-Subtract Rule: AN

AK = AN – K

We write the quantity A multiplied to itself N times as AN, i.e.

A x A x A ….x A = AN

Example D . 56

52 = (5)(5)(5)(5)(5)(5)(5)(5)

Example B.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2

xy2 = (x)(yy) –x2 = –(xx)

base

exponent

Exponents

Multiply-Add Rule: ANAK =AN+K Example C .

a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57

b. x5y7x4y6 = x5x4y7y6 = x9y13

Rules of Exponents

Divide-Subtract Rule: AN

AK = AN – K

We write the quantity A multiplied to itself N times as AN, i.e.

A x A x A ….x A = AN

Example D . 56

52 = (5)(5)(5)(5)(5)(5)(5)(5) = 56 – 2

Example B.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2

xy2 = (x)(yy) –x2 = –(xx)

base

exponent

Exponents

Multiply-Add Rule: ANAK =AN+K Example C .

a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57

b. x5y7x4y6 = x5x4y7y6 = x9y13

Rules of Exponents

Divide-Subtract Rule: AN

AK = AN – K

We write the quantity A multiplied to itself N times as AN, i.e.

A x A x A ….x A = AN

Example D . 56

52 = (5)(5)(5)(5)(5)(5)(5)(5) = 56 – 2 = 54

Power-Multiply Rule: (AN)K = ANK

Example E. (34)5 = (34)(34)(34)(34)(34)

= 34+4+4+4+4

= 34*5 = 320

Exponents

Power-Multiply Rule: (AN)K = ANK

Example E. (34)5 = (34)(34)(34)(34)(34)

= 34+4+4+4+4

= 34*5 = 320

Exponents

Power-Distribute Rule: (ANBM)K = ANK BMK

Example F. (34 23)5 = (34 23)(34 23)(34 23)(34 23)(34 23)

= 34+4+4+4+4 2 3+3+3+3+3 = 34*5 23*5 = 320 215

Power-Multiply Rule: (AN)K = ANK

Example E. (34)5 = (34)(34)(34)(34)(34)

= 34+4+4+4+4

= 34*5 = 320

Exponents

Power-Distribute Rule: (ANBM)K = ANK BMK

Example F. (34 23)5 = (34 23)(34 23)(34 23)(34 23)(34 23)

= 34+4+4+4+4 2 3+3+3+3+3 = 34*5 23*5 = 320 215!Note: (AN ± BM)K = ANK ± BMK, e.g. (2 + 3)2 = 22 + 32.

Power-Multiply Rule: (AN)K = ANK

Example E. (34)5 = (34)(34)(34)(34)(34)

= 34+4+4+4+4

= 34*5 = 320

Exponents

Power-Distribute Rule: (ANBM)K = ANK BMK

Example F. (34 23)5 = (34 23)(34 23)(34 23)(34 23)(34 23)

= 34+4+4+4+4 2 3+3+3+3+3 = 34*5 23*5 = 320 215

The positive–whole–number exponent specifies a tangible number of copies of the base to be multiplied (e.g. A2 = A x A, 2 copies of A). Let’s extend exponent notation to other types of exponents such as A0 or A–1. However A0 does not mean there is “0” copy of A, or that A–1 is “–1” copy of A.

Non–Positive–Whole–Number Exponents !Note: (AN ± BM)K = ANK ± BMK, e.g. (2 + 3)2 = 22 + 32.

ExponentsWe extract the meaning of A0 or A–1 by examining the consequences of the above rules.

Exponents

Since = 1A1

A1

We extract the meaning of A0 or A–1 by examining the consequences of the above rules.

Exponents

Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A1

A1

We extract the meaning of A0 or A–1 by examining the consequences of the above rules.

Exponents

Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A1

A1

0-Power Rule: A0 = 1, A = 0

We extract the meaning of A0 or A–1 by examining the consequences of the above rules.

Exponents

Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A1

A1

Since =1A

A0

A1

0-Power Rule: A0 = 1, A = 0

We extract the meaning of A0 or A–1 by examining the consequences of the above rules.

Exponents

Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A1

A1

Since = = A0 – 1 = A–1, we get the negative-power rule.1A

A0

A1

0-Power Rule: A0 = 1, A = 0

We extract the meaning of A0 or A–1 by examining the consequences of the above rules.

Exponents

Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A1

A1

Since = = A0 – 1 = A–1, we get the negative-power rule.1A

A0

A1

Negative-Power Rule: A–1 = 1A

0-Power Rule: A0 = 1, A = 0

, A = 0

We extract the meaning of A0 or A–1 by examining the consequences of the above rules.

Exponents

Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A1

A1

Since = = A0 – 1 = A–1, we get the negative-power rule.1A

A0

A1

Negative-Power Rule: A–1 = 1A

0-Power Rule: A0 = 1, A = 0

, A = 0

We extract the meaning of A0 or A–1 by examining the consequences of the above rules.

and in general that1AKA–K =

Exponents

Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A1

A1

Since = = A0 – 1 = A–1, we get the negative-power rule.1A

A0

A1

Negative-Power Rule: A–1 = 1A

0-Power Rule: A0 = 1, A = 0

, A = 0

We extract the meaning of A0 or A–1 by examining the consequences of the above rules.

and in general that1AK

The “negative” of an exponents mean to reciprocate the base. A–K =

Exponents

Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A1

A1

Since = = A0 – 1 = A–1, we get the negative-power rule.1A

A0

A1

Negative-Power Rule: A–1 = 1A

Example J. Simplify

c. ( )–1 2 5 =

b. 3–2 =a. 30 =

0-Power Rule: A0 = 1, A = 0

, A = 0

We extract the meaning of A0 or A–1 by examining the consequences of the above rules.

and in general that1AK

The “negative” of an exponents mean to reciprocate the base. A–K =

Exponents

Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A1

A1

Since = = A0 – 1 = A–1, we get the negative-power rule.1A

A0

A1

Negative-Power Rule: A–1 = 1A

Example J. Simplify

c. ( )–1 2 5

b. 3–2 =a. 30 = 1

0-Power Rule: A0 = 1, A = 0

, A = 0

We extract the meaning of A0 or A–1 by examining the consequences of the above rules.

and in general that1AK

The “negative” of an exponents mean to reciprocate the base. A–K =

=

Exponents

Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A1

A1

Since = = A0 – 1 = A–1, we get the negative-power rule.1A

A0

A1

Negative-Power Rule: A–1 = 1A

Example J. Simplify

1 32

1 9

c. ( )–1 2 5 =

b. 3–2 = =a. 30 = 1

0-Power Rule: A0 = 1, A = 0

, A = 0

We extract the meaning of A0 or A–1 by examining the consequences of the above rules.

and in general that1AK

The “negative” of an exponents mean to reciprocate the base. A–K =

Exponents

Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A1

A1

Since = = A0 – 1 = A–1, we get the negative-power rule.1A

A0

A1

Negative-Power Rule: A–1 = 1A

Example J. Simplify

1 32

1 9

c. ( )–1 2 5 = 1

2/5 = 1* 5 2 = 5

2

b. 3–2 = =a. 30 = 1

0-Power Rule: A0 = 1, A = 0

, A = 0

We extract the meaning of A0 or A–1 by examining the consequences of the above rules.

and in general that1AK

The “negative” of an exponents mean to reciprocate the base. A–K =

Exponents

Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A1

A1

Since = = A0 – 1 = A–1, we get the negative-power rule.1A

A0

A1

Negative-Power Rule: A–1 = 1A

Example J. Simplify

1 32

1 9

c. ( )–1 2 5 = 1

2/5 = 1* 5 2 = 5

2

b. 3–2 = =a. 30 = 1

In general ( )–K

a b = ( )K b

a d. ( )–2 2

5 =

0-Power Rule: A0 = 1, A = 0

, A = 0

We extract the meaning of A0 or A–1 by examining the consequences of the above rules.

and in general that1AK

The “negative” of an exponents mean to reciprocate the base. A–K =

Exponents

Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A1

A1

Since = = A0 – 1 = A–1, we get the negative-power rule.1A

A0

A1

Negative-Power Rule: A–1 = 1A

Example J. Simplify

1 32

1 9

c. ( )–1 2 5 = 1

2/5 = 1* 5 2 = 5

2

b. 3–2 = =a. 30 = 1

In general ( )–K

a b = ( )K b

a d. ( )–2 2

5 = ( )2 = 25 4

5 2

0-Power Rule: A0 = 1, A = 0

, A = 0

We extract the meaning of A0 or A–1 by examining the consequences of the above rules.

and in general that1AK

The “negative” of an exponents mean to reciprocate the base. A–K =

e. 3–1 – 40 * 2–2 =

Exponents

e. 3–1 – 40 * 2–2 = 1 3

Exponents

e. 3–1 – 40 * 2–2 = 1 3 – 1*

Exponents

e. 3–1 – 40 * 2–2 = 1 3 – 1* 1

22

Exponents

e. 3–1 – 40 * 2–2 = 1 3 – 1* 1

22 = 1 3

– 1 4 = 1

12

Exponents

e. 3–1 – 40 * 2–2 = 1 3 – 1* 1

22 = 1 3

– 1 4 = 1

12

Exponents

Although the negative power means to reciprocate, for problems of consolidating exponents, we do not reciprocate the negative exponents.

e. 3–1 – 40 * 2–2 = 1 3 – 1* 1

22 = 1 3

– 1 4 = 1

12

Exponents

Although the negative power means to reciprocate, for problems of consolidating exponents, we do not reciprocate the negative exponents. Instead we add or subtract them using the multiplication and division rules first.

e. 3–1 – 40 * 2–2 =

Exponents

Although the negative power means to reciprocate, for problems of consolidating exponents, we do not reciprocate the negative exponents. Instead we add or subtract them using the multiplication and division rules first.Example H. Simplify 3–2 x4 y–6 x–8 y 23

1 3 – 1* 1

22 = 1 3

– 1 4 = 1

12

e. 3–1 – 40 * 2–2 =

Exponents

Although the negative power means to reciprocate, for problems of consolidating exponents, we do not reciprocate the negative exponents. Instead we add or subtract them using the multiplication and division rules first.Example H. Simplify 3–2 x4 y–6 x–8 y 23

3–2 x4 y–6 x–8 y23

1 3 – 1* 1

22 = 1 3

– 1 4 = 1

12

e. 3–1 – 40 * 2–2 =

Exponents

Although the negative power means to reciprocate, for problems of consolidating exponents, we do not reciprocate the negative exponents. Instead we add or subtract them using the multiplication and division rules first.Example H. Simplify 3–2 x4 y–6 x–8 y 23

3–2 x4 y–6 x–8 y23

= 3–2 x4 x–8 y–6 y23

1 3 – 1* 1

22 = 1 3

– 1 4 = 1

12

e. 3–1 – 40 * 2–2 =

Exponents

Although the negative power means to reciprocate, for problems of consolidating exponents, we do not reciprocate the negative exponents. Instead we add or subtract them using the multiplication and division rules first.

= x4 – 8 y–6+23

Example H. Simplify 3–2 x4 y–6 x–8 y 23

3–2 x4 y–6 x–8 y23

= 3–2 x4 x–8 y–6 y23

1 9

1 3 – 1* 1

22 = 1 3

– 1 4 = 1

12

e. 3–1 – 40 * 2–2 =

Exponents

Although the negative power means to reciprocate, for problems of consolidating exponents, we do not reciprocate the negative exponents. Instead we add or subtract them using the multiplication and division rules first.

= x4 – 8 y–6+23

= x–4 y17

Example H. Simplify 3–2 x4 y–6 x–8 y 23

3–2 x4 y–6 x–8 y23

= 3–2 x4 x–8 y–6 y23

1 9 1 9

1 3 – 1* 1

22 = 1 3

– 1 4 = 1

12

e. 3–1 – 40 * 2–2 =

Exponents

Although the negative power means to reciprocate, for problems of consolidating exponents, we do not reciprocate the negative exponents. Instead we add or subtract them using the multiplication and division rules first.

= x4 – 8 y–6+23

= x–4 y17

= y17

Example H. Simplify 3–2 x4 y–6 x–8 y 23

3–2 x4 y–6 x–8 y23

= 3–2 x4 x–8 y–6 y23

1 9 1 9

1 9x4

1 3 – 1* 1

22 = 1 3

– 1 4 = 1

12

e. 3–1 – 40 * 2–2 =

Exponents

Although the negative power means to reciprocate, for problems of consolidating exponents, we do not reciprocate the negative exponents. Instead we add or subtract them using the multiplication and division rules first.

= x4 – 8 y–6+23

= x–4 y17

= y17

=

Example H. Simplify 3–2 x4 y–6 x–8 y 23

3–2 x4 y–6 x–8 y23

= 3–2 x4 x–8 y–6 y23

1 9 1 9

1 9x4

y17

9x4

1 3 – 1* 1

22 = 1 3

– 1 4 = 1

12

ExponentsExample I. Simplify using the rules for exponents. Leave the answer in positive exponents only.

23x–8

26 x–3

ExponentsExample I. Simplify using the rules for exponents. Leave the answer in positive exponents only.

23x–8

26 x–3

23x–8

26x–3

ExponentsExample I. Simplify using the rules for exponents. Leave the answer in positive exponents only.

23x–8

26 x–3

23x–8

26x–3 = 23 – 6 x–8 – (–3 )

ExponentsExample I. Simplify using the rules for exponents. Leave the answer in positive exponents only.

23x–8

26 x–3

23x–8

26x–3 = 23 – 6 x–8 – (–3 )

= 2–3 x–5

ExponentsExample I. Simplify using the rules for exponents. Leave the answer in positive exponents only.

23x–8

26 x–3

23x–8

26x–3 = 23 – 6 x–8 – (–3 )

= 2–3 x–5

= 231

x51

* = 8x51

ExponentsExample I. Simplify using the rules for exponents. Leave the answer in positive exponents only.

23x–8

26 x–3

23x–8

26x–3 = 23 – 6 x–8 – (–3 )

= 2–3 x–5

= 231

x51

* = 8x51

Example J. Simplify (3x–2y3)–2 x2

3–5x–3(y–1x2)3

ExponentsExample I. Simplify using the rules for exponents. Leave the answer in positive exponents only.

23x–8

26 x–3

23x–8

26x–3 = 23 – 6 x–8 – (–3 )

= 2–3 x–5

= 231

x51

* = 8x51

Example J. Simplify (3x–2y3)–2 x2

3–5x–3(y–1x2)3

(3x–2y3)–2 x2

3–5x–3(y–1x2)3

ExponentsExample I. Simplify using the rules for exponents. Leave the answer in positive exponents only.

23x–8

26 x–3

23x–8

26x–3 = 23 – 6 x–8 – (–3 )

= 2–3 x–5

= 231

x51

* = 8x51

Example J. Simplify (3x–2y3)–2 x2

3–5x–3(y–1x2)3

= 3–2x4y–6x2

3–5x–3y–3 x6 (3x–2y3)–2 x2

3–5x–3(y–1x2)3

ExponentsExample I. Simplify using the rules for exponents. Leave the answer in positive exponents only.

23x–8

26 x–3

23x–8

26x–3 = 23 – 6 x–8 – (–3 )

= 2–3 x–5

= 231

x51

* = 8x51

Example J. Simplify (3x–2y3)–2 x2

3–5x–3(y–1x2)3

= 3–2x4y–6x2

3–5x–3y–3 x6 =(3x–2y3)–2 x2

3–5x–3(y–1x2)3 3–5x–3x6y–3 3–2x4x2y–6

ExponentsExample I. Simplify using the rules for exponents. Leave the answer in positive exponents only.

23x–8

26 x–3

23x–8

26x–3 = 23 – 6 x–8 – (–3 )

= 2–3 x–5

= 231

x51

* = 8x51

Example J. Simplify (3x–2y3)–2 x2

3–5x–3(y–1x2)3

= 3–2x4y–6x2

3–5x–3y–3 x6 =

=

(3x–2y3)–2 x2

3–5x–3(y–1x2)3 3–5x–3x6y–3 3–2x4x2y–6

3–2x6y–6

3–5x3y–3

ExponentsExample I. Simplify using the rules for exponents. Leave the answer in positive exponents only.

23x–8

26 x–3

23x–8

26x–3 = 23 – 6 x–8 – (–3 )

= 2–3 x–5

= 231

x51

* = 8x51

Example J. Simplify (3x–2y3)–2 x2

3–5x–3(y–1x2)3

= 3–2x4y–6x2

3–5x–3y–3 x6 =

= = 3–2 – (–5) x6 – 3 y–6 – (–3)

(3x–2y3)–2 x2

3–5x–3(y–1x2)3 3–5x–3x6y–3 3–2x4x2y–6

3–2x6y–6

3–5x3y–3

ExponentsExample I. Simplify using the rules for exponents. Leave the answer in positive exponents only.

23x–8

26 x–3

23x–8

26x–3 = 23 – 6 x–8 – (–3 )

= 2–3 x–5

= 231

x51

* = 8x51

Example J. Simplify (3x–2y3)–2 x2

3–5x–3(y–1x2)3

= 3–2x4y–6x2

3–5x–3y–3 x6 =

= = 3–2 – (–5) x6 – 3 y–6 – (–3)

= 33 x3 y–3 =

(3x–2y3)–2 x2

3–5x–3(y–1x2)3 3–5x–3x6y–3 3–2x4x2y–6

3–2x6y–6

3–5x3y–3

ExponentsExample I. Simplify using the rules for exponents. Leave the answer in positive exponents only.

23x–8

26 x–3

23x–8

26x–3 = 23 – 6 x–8 – (–3 )

= 2–3 x–5

= 231

x51

* = 8x51

Example J. Simplify (3x–2y3)–2 x2

3–5x–3(y–1x2)3

= 3–2x4y–6x2

3–5x–3y–3 x6 =

= = 3–2 – (–5) x6 – 3 y–6 – (–3)

= 33 x3 y–3 = 27 x3

(3x–2y3)–2 x2

3–5x–3(y–1x2)3 3–5x–3x6y–3 3–2x4x2y–6

3–2x6y–6

3–5x3y–3

y3

ExponentsExample I. Simplify using the rules for exponents. Leave the answer in positive exponents only.

23x–8

26 x–3

23x–8

26x–3 = 23 – 6 x–8 – (–3 )

= 2–3 x–5

= 231

x51

* = 8x51

Example J. Simplify (3x–2y3)–2 x2

3–5x–3(y–1x2)3

= 3–2x4y–6x2

3–5x–3y–3 x6 =

= = 3–2 – (–5) x6 – 3 y–6 – (–3)

= 33 x3 y–3 = 27 x3

(3x–2y3)–2 x2

3–5x–3(y–1x2)3 3–5x–3x6y–3 3–2x4x2y–6

3–2x6y–6

3–5x3y–3

y3