Exponents

Back to Algebra–Ready Review Content.

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-Power 1: 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 B. Calculate the following.

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

The Blank-Power 1: 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 B. Calculate the following.

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

The Blank-Power 1: 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

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 B. Calculate the following.

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

The Blank-Power 1: 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 B. Calculate the following.

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

The Blank-Power 1: 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 B. Calculate the following.

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

The Blank-Power 1: 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 B. Calculate the following.

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

The Blank-Power 1: 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 B. Calculate the following.

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

The Blank-Power 1: 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

Example B. Calculate the following.

= 2*2*3 = 2*3*3

= 6*3

= 18

= 2*2*3*3*3

d. 22 x 3 e. 2 x 32 f. 22 x 33

= 36 * 3

= 4 * 9= 12 * 3

= 108

Exponents

Example B. Calculate the following.

= 2*2*3 = 2*3*3

= 6*3

= 18

= 2*2*3*3*3

d. 22 x 3 e. 2 x 32 f. 22 x 33

= 36 * 3

= 4 * 9= 12 * 3

= 108

These problems are the same as 22(3), 2(32), and 22(33).About the Notation

Recall that for addition, we write 2 + 2 + 2 as 3 x 2 = 3(2) = 2(3) 3 copies

2 * 2 * 2 = 23

3 copies

For repetitive multiplication,

to distinguish it from addition, we store the 3 in the upper corner as 23.

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 A.

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 A.

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 A.

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 A.

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 A.

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 A.

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 A.

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 A.

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 A.

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 A.

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 A.

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 A.

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 B.

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 A.

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 B.

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 A.

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 B.

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 A.

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 B.

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 A.

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 B.

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 A.

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 B.

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 C. 56

52 =

Example A.

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 B.

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 C. 56

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

Example A.

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 B.

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 C. 56

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

Example A.

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 B.

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 C. 56

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

Power-Multiply Rule : (AN)K = ANK

Example D. (34)5 =

Exponents

Power-Multiply Rule : (AN)K = ANK

Example D. (34)5 = (34)(34)(34)(34)(34)

Exponents

Power-Multiply Rule : (AN)K = ANK

Example D. (34)5 = (34)(34)(34)(34)(34)

= 34+4+4+4+4

Exponents

Power-Multiply Rule : (AN)K = ANK

Example D. (34)5 = (34)(34)(34)(34)(34)

= 34+4+4+4+4

= 34*5 = 320

Exponents

Power-Multiply Rule : (AN)K = ANK

Example D. (34)5 = (34)(34)(34)(34)(34)

= 34+4+4+4+4

= 34*5 = 320

Exponents

Since = 1 = A1 – 1A1

A1

Power-Multiply Rule : (AN)K = ANK

Example D. (34)5 = (34)(34)(34)(34)(34)

= 34+4+4+4+4

= 34*5 = 320

Exponents

Since = 1 = A1 – 1 = A0, A1

A1

Power-Multiply Rule : (AN)K = ANK

Example D. (34)5 = (34)(34)(34)(34)(34)

= 34+4+4+4+4

= 34*5 = 320

Exponents

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

A1

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

Power-Multiply Rule : (AN)K = ANK

Example D. (34)5 = (34)(34)(34)(34)(34)

= 34+4+4+4+4

= 34*5 = 320

Exponents

An important application for exponents is the use of the powers of 10.

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

A1

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

Power-Multiply Rule : (AN)K = ANK

Example D. (34)5 = (34)(34)(34)(34)(34)

= 34+4+4+4+4

= 34*5 = 320

Exponents

100 = 1 101 = 10 102 = 100 103 = 1,000

An important application for exponents is the use of the powers of 10. In particular,

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

A1

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

onetenone hundredone thousand

106 = 1,000,000 one million

109 = 1,000,000,000 one billion

1012 = 1,000,000,000,000 one trillion

Power-Multiply Rule : (AN)K = ANK

Example D. (34)5 = (34)(34)(34)(34)(34)

= 34+4+4+4+4

= 34*5 = 320

Exponents

100 = 1 101 = 10 102 = 100 103 = 1,000

An important application for exponents is the use of the powers of 10. In particular,

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

A1

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

onetenone hundredone thousand

106 = 1,000,000 one million

109 = 1,000,000,000 one billion

1012 = 1,000,000,000,000 one trillion

For example, the national debt of USA, as of Dec. 2014, is 18 trillions dollars or $18,000,000,000,000 (12 0’s) may be written as 18*1012 which is easier to read.