Exponents

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

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

A x A x A ….x A = AN

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

Multiplication 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

Multiplication 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

Multiplication 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

Multiplication 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

Multiplication 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

Multiplication 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

Multiplication 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

Division 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 A.

43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2

xy2 = (x)(yy) –x2 = –(xx)

base

exponent

Exponents

Multiplication 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

Division Rule:

Example C.

AN

AK = AN – K

56

52

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

Multiplication 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

Division Rule:

Example C.

AN

AK = AN – K

56

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

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

Multiplication 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

Division Rule:

Example C.

AN

AK = AN – K

56

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

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

Multiplication 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

Division Rule:

Example C.

AN

AK = AN – K

56

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

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

A x A x A ….x A = AN

Power Rule: (AN)K = ANK

Exponents

Power Rule: (AN)K = ANK

Example D. (34)5

Exponents

A1

Power Rule: (AN)K = ANK

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

Exponents

Power Rule: (AN)K = ANK

Example D. (34)5 = (34)(34)(34)(34)(34) = 34+4+4+4+4

Exponents

Power Rule: (AN)K = ANK

Example D. (34)5 = (34)(34)(34)(34)(34) = 34+4+4+4+4 = 34*5 = 320

Exponents

Power 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

A1

Power 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 A1

A1

Power 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 = A0A1

A1

Power 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

Power 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

Power 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

Since = 1AK

A0

AK

Power 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

Since = = A0 – K 1AK

A0

AK

Power 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

Since = = A0 – K = A–K, we get the negative-power Rule.1AK

A0

AK

Power 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

Since = = A0 – K = A–K, we get the negative-power Rule.1AK

A0

AK

Negative-Power Rule: A–K = 1AK

Power 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

Since = = A0 – K = A–K, we get the negative-power Rule.1AK

A0

AK

Negative-Power Rule: A–K = 1AK

Example E. Simplify

a. 30

Power 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

Since = = A0 – K = A–K, we get the negative-power Rule.1AK

A0

AK

Negative-Power Rule: A–K = 1AK

Example E. Simplify

a. 30 = 1

Power 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

Since = = A0 – K = A–K, we get the negative-power Rule.1AK

A0

AK

Negative-Power Rule: A–K = 1AK

Example E. Simplify

b. 3–2

a. 30 = 1

Power 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

Since = = A0 – K = A–K, we get the negative-power Rule.1AK

A0

AK

Negative-Power Rule: A–K = 1AK

Example E. Simplify

1 32 b. 3–2 =

a. 30 = 1

Power 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

Since = = A0 – K = A–K, we get the negative-power Rule.1AK

A0

AK

Negative-Power Rule: A–K = 1AK

Example E. Simplify

1 32

1 9b. 3–2 = =

a. 30 = 1

Power 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

Since = = A0 – K = A–K, we get the negative-power Rule.1AK

A0

AK

Negative-Power Rule: A–K = 1AK

Example E. Simplify

1 32

1 9

c. ( )–1 2 5

b. 3–2 = =a. 30 = 1

Power 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

Since = = A0 – K = A–K, we get the negative-power Rule.1AK

A0

AK

Negative-Power Rule: A–K = 1AK

Example E. Simplify

1 32

1 9

c. ( )–1 2 5 = 1

2/5 =

b. 3–2 = =a. 30 = 1

Power 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

Since = = A0 – K = A–K, we get the negative-power Rule.1AK

A0

AK

Negative-Power Rule: A–K = 1AK

Example E. Simplify

1 32

1 9

c. ( )–1 2 5 = 1

2/5 = 1* 5 2 = 5

2

b. 3–2 = =a. 30 = 1

Power 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

Since = = A0 – K = A–K, we get the negative-power Rule.1AK

A0

AK

Negative-Power Rule: A–K = 1AK

Example E. 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

Power 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

Since = = A0 – K = A–K, we get the negative-power Rule.1AK

A0

AK

Negative-Power Rule: A–K = 1AK

Example E. 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

Power 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

Since = = A0 – K = A–K, we get the negative-power Rule.1AK

A0

AK

Negative-Power Rule: A–K = 1AK

Example E. 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 5

2

Power 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

Since = = A0 – K = A–K, we get the negative-power Rule.1AK

A0

AK

Negative-Power Rule: A–K = 1AK

Example E. 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

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 collecting 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 collecting 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 collecting exponents, we do not reciprocate the negative exponents. Instead we add or subtract them using the multiplication and division rules first.

Example F. 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 collecting exponents, we do not reciprocate the negative exponents. Instead we add or subtract them using the multiplication and division rules first.

Example F. 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 collecting exponents, we do not reciprocate the negative exponents. Instead we add or subtract them using the multiplication and division rules first.

Example F. 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 collecting 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 F. 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 collecting 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 F. 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 collecting 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 F. 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 collecting 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 F. 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 G. Simplify using the rules for exponents.

Leave the answer in positive exponents only.

23x–8

26 x–3

ExponentsExample G. Simplify using the rules for exponents.

Leave the answer in positive exponents only.

23x–8

26 x–3

23x–8

26x–3

ExponentsExample G. 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 G. 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 G. 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 G. 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 H. Simplify (3x–2y3)–2 x2

3–5x–3(y–1x2)3

ExponentsExample G. 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 H. Simplify (3x–2y3)–2 x2

3–5x–3(y–1x2)3

(3x–2y3)–2 x2

3–5x–3(y–1x2)3

ExponentsExample G. 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 H. 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 G. 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 H. 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 G. 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 H. 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 G. 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 H. 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 G. 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 H. 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 G. 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 H. 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 G. 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 H. 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

An important application of exponents is the scientific notation.

Scientific Notation

Scientific NotationScientific notation simplifies the tracking and calculation of very large or very small numbers. We note the relation between the exponents and the base-10 numbers:

100 = 1

Scientific NotationScientific notation simplifies the tracking and calculation of very large or very small numbers. We note the relation between the exponents and the base-10 numbers:

100 = 1 101 = 10

Scientific NotationScientific notation simplifies the tracking and calculation of very large or very small numbers. We note the relation between the exponents and the base-10 numbers:

100 = 1 101 = 10 102 = 100

Scientific NotationScientific notation simplifies the tracking and calculation of very large or very small numbers. We note the relation between the exponents and the base-10 numbers:

100 = 1 101 = 10 102 = 100 103 = 1000

Scientific NotationScientific notation simplifies the tracking and calculation of very large or very small numbers. We note the relation between the exponents and the base-10 numbers:

100 = 1 101 = 10 102 = 100 103 = 1000

10–1 = 0.1

Scientific NotationScientific notation simplifies the tracking and calculation of very large or very small numbers. We note the relation between the exponents and the base-10 numbers:

100 = 1 101 = 10 102 = 100 103 = 1000

10–1 = 0.1 10–2 = 0.01

Scientific NotationScientific notation simplifies the tracking and calculation of very large or very small numbers. We note the relation between the exponents and the base-10 numbers:

100 = 1 101 = 10 102 = 100 103 = 1000

10–1 = 0.1 10–2 = 0.01 10–3 = 0.001 10–4 = 0.0001

Scientific NotationScientific notation simplifies the tracking and calculation of very large or very small numbers. We note the relation between the exponents and the base-10 numbers:

100 = 1 101 = 10 102 = 100 103 = 1000

10–1 = 0.1 10–2 = 0.01 10–3 = 0.001 10–4 = 0.0001

Scientific Notation

Scientific Notation

Scientific notation simplifies the tracking and calculation of very large or very small numbers. We note the relation between the exponents and the base-10 numbers:

100 = 1 101 = 10 102 = 100 103 = 1000

10–1 = 0.1 10–2 = 0.01 10–3 = 0.001 10–4 = 0.0001

Scientific Notation

Scientific NotationAny number can be written in the form

A x 10N

where 1 < A < 10.

Scientific notation simplifies the tracking and calculation of very large or very small numbers. We note the relation between the exponents and the base-10 numbers:

100 = 1 101 = 10 102 = 100 103 = 1000

10–1 = 0.1 10–2 = 0.01 10–3 = 0.001 10–4 = 0.0001

Scientific Notation

Scientific NotationAny number can be written in the form

A x 10N

where 1 < A < 10. This form is called the scientific notation of the number.

Scientific notation simplifies the tracking and calculation of very large or very small numbers. We note the relation between the exponents and the base-10 numbers:

Scientific NotationTo write a number in scientific notation, we move the decimal point behind the first nonzero digit.

Scientific NotationTo write a number in scientific notation, we move the decimal point behind the first nonzero digit.i. If the decimal point moves to the left N spaces, then the exponent over 10 is positive N,

Scientific NotationTo write a number in scientific notation, we move the decimal point behind the first nonzero digit.i. If the decimal point moves to the left N spaces, then the exponent over 10 is positive N, i.e. if after moving the decimal point we get a smaller number A, then N is positive.

Scientific NotationTo write a number in scientific notation, we move the decimal point behind the first nonzero digit.i. If the decimal point moves to the left N spaces, then the exponent over 10 is positive N, i.e. if after moving the decimal point we get a smaller number A, then N is positive.

Example I. Write the following numbers in scientific notation.

a. 12300.

Scientific NotationTo write a number in scientific notation, we move the decimal point behind the first nonzero digit.i. If the decimal point moves to the left N spaces, then the exponent over 10 is positive N, i.e. if after moving the decimal point we get a smaller number A, then N is positive.

Move left 4 places.

Example I. Write the following numbers in scientific notation.

a. 12300. = 1 2300 .

Scientific NotationTo write a number in scientific notation, we move the decimal point behind the first nonzero digit.i. If the decimal point moves to the left N spaces, then the exponent over 10 is positive N, i.e. if after moving the decimal point we get a smaller number A, then N is positive.

Move left 4 places.

Example I. Write the following numbers in scientific notation.

a. 12300. = 1 2300 . = 1. 23 x 10 +4

Scientific NotationTo write a number in scientific notation, we move the decimal point behind the first nonzero digit.i. If the decimal point moves to the left N spaces, then the exponent over 10 is positive N, i.e. if after moving the decimal point we get a smaller number A, then N is positive.ii. If the decimal point moves to the right N spaces, then the exponent over 10 is negative,

Move left 4 places.

Example I. Write the following numbers in scientific notation.

a. 12300. = 1 2300 . = 1. 23 x 10 +4

Scientific NotationTo write a number in scientific notation, we move the decimal point behind the first nonzero digit.i. If the decimal point moves to the left N spaces, then the exponent over 10 is positive N, i.e. if after moving the decimal point we get a smaller number A, then N is positive.ii. If the decimal point moves to the right N spaces, then the exponent over 10 is negative, i.e. if after moving the decimal point we get a larger number A, then N is negative.

Move left 4 places.

Example I. Write the following numbers in scientific notation.

a. 12300. = 1 2300 . = 1. 23 x 10 +4

Scientific NotationTo write a number in scientific notation, we move the decimal point behind the first nonzero digit.i. If the decimal point moves to the left N spaces, then the exponent over 10 is positive N, i.e. if after moving the decimal point we get a smaller number A, then N is positive.ii. If the decimal point moves to the right N spaces, then the exponent over 10 is negative, i.e. if after moving the decimal point we get a larger number A, then N is negative.

Move left 4 places.

Example I. Write the following numbers in scientific notation.

a. 12300. = 1 2300 . = 1. 23 x 10 +4

b. 0.00123

Scientific NotationTo write a number in scientific notation, we move the decimal point behind the first nonzero digit.i. If the decimal point moves to the left N spaces, then the exponent over 10 is positive N, i.e. if after moving the decimal point we get a smaller number A, then N is positive.ii. If the decimal point moves to the right N spaces, then the exponent over 10 is negative, i.e. if after moving the decimal point we get a larger number A, then N is negative.

Move left 4 places.

Move right 3 places

Example I. Write the following numbers in scientific notation.

a. 12300. = 1 2300 . = 1. 23 x 10 +4

b. 0.00123 = 0. 001 23

Scientific NotationTo write a number in scientific notation, we move the decimal point behind the first nonzero digit.i. If the decimal point moves to the left N spaces, then the exponent over 10 is positive N, i.e. if after moving the decimal point we get a smaller number A, then N is positive.ii. If the decimal point moves to the right N spaces, then the exponent over 10 is negative, i.e. if after moving the decimal point we get a larger number A, then N is negative.

Move left 4 places.

Move right 3 places

Example I. Write the following numbers in scientific notation.

a. 12300. = 1 2300 . = 1. 23 x 10 +4

b. 0.00123 = 0. 001 23 = 1. 23 x 10 –3

Scientific NotationTo change a number in scientific notation back to the standard form, we move the decimal point according to N.

Scientific NotationTo change a number in scientific notation back to the standard form, we move the decimal point according to N.i. If N is positive, move the decimal point in A to the right,

Scientific NotationTo change a number in scientific notation back to the standard form, we move the decimal point according to N.i. If N is positive, move the decimal point in A to the right, i.e. make A into a larger number.

Scientific NotationTo change a number in scientific notation back to the standard form, we move the decimal point according to N.i. If N is positive, move the decimal point in A to the right, i.e. make A into a larger number.

Example J. Write the following numbers in the standard form.

a. 1. 23 x 10 +4

Scientific NotationTo change a number in scientific notation back to the standard form, we move the decimal point according to N.i. If N is positive, move the decimal point in A to the right, i.e. make A into a larger number.

Move right 4 places,

Example J. Write the following numbers in the standard form.

a. 1. 23 x 10 +4 = 1 2300 . = 12300.

Scientific NotationTo change a number in scientific notation back to the standard form, we move the decimal point according to N.i. If N is positive, move the decimal point in A to the right, i.e. make A into a larger number.ii. If N is negative, move the decimal point in A to the left,

Move right 4 places,

Example J. Write the following numbers in the standard form.

a. 1. 23 x 10 +4 = 1 2300 . = 12300.

Scientific NotationTo change a number in scientific notation back to the standard form, we move the decimal point according to N.i. If N is positive, move the decimal point in A to the right, i.e. make A into a larger number.ii. If N is negative, move the decimal point in A to the left, i.e. make A into a smaller number.

Move right 4 places,

Example J. Write the following numbers in the standard form.

a. 1. 23 x 10 +4 = 1 2300 . = 12300.

Scientific NotationTo change a number in scientific notation back to the standard form, we move the decimal point according to N.i. If N is positive, move the decimal point in A to the right, i.e. make A into a larger number.ii. If N is negative, move the decimal point in A to the left, i.e. make A into a smaller number.

Move right 4 places,

Example J. Write the following numbers in the standard form.

a. 1. 23 x 10 +4 = 1 2300 . = 12300.

b. 1. 23 x 10 –3

Scientific NotationTo change a number in scientific notation back to the standard form, we move the decimal point according to N.i. If N is positive, move the decimal point in A to the right, i.e. make A into a larger number.ii. If N is negative, move the decimal point in A to the left, i.e. make A into a smaller number.

Move right 4 places,

Move left 3 places

Example J. Write the following numbers in the standard form.

a. 1. 23 x 10 +4 = 1 2300 . = 12300.

b. 1. 23 x 10 –3 = 0. 001 23 = 0.00123

Scientific NotationTo change a number in scientific notation back to the standard form, we move the decimal point according to N.i. If N is positive, move the decimal point in A to the right, i.e. make A into a larger number.ii. If N is negative, move the decimal point in A to the left, i.e. make A into a smaller number.

Move right 4 places,

Move left 3 places

Example J. Write the following numbers in the standard form.

a. 1. 23 x 10 +4 = 1 2300 . = 12300.

b. 1. 23 x 10 –3 = 0. 001 23 = 0.00123

Scientific notation simplifies multiplication and division of very large and very small numbers.

Example K. Calculate. Give the answer in both scientific notation and the standard notation.

a. (1.2 x 108) x (1.3 x 10–12)

Scientific Notation

Example K. Calculate. Give the answer in both scientific notation and the standard notation.

a. (1.2 x 108) x (1.3 x 10–12) = 1.2 x 1.3 x 108 x 10 –12

Scientific Notation

Example K. Calculate. Give the answer in both scientific notation and the standard notation.

a. (1.2 x 108) x (1.3 x 10–12) = 1.2 x 1.3 x 108 x 10 –12

= 1.56 x 108 –12

Scientific Notation

Example K. Calculate. Give the answer in both scientific notation and the standard notation.

a. (1.2 x 108) x (1.3 x 10–12) = 1.2 x 1.3 x 108 x 10 –12

= 1.56 x 108 –12

= 1.56 x 10 –4

Scientific Notation

Example K. Calculate. Give the answer in both scientific notation and the standard notation.

a. (1.2 x 108) x (1.3 x 10–12) = 1.2 x 1.3 x 108 x 10 –12

= 1.56 x 108 –12

= 1.56 x 10 –4 = 0.000156

Scientific Notation

Example K. Calculate. Give the answer in both scientific notation and the standard notation.

a. (1.2 x 108) x (1.3 x 10–12) = 1.2 x 1.3 x 108 x 10 –12

= 1.56 x 108 –12

= 1.56 x 10 –4 = 0.000156

b. 6.3 x 10-2 2.1 x 10-10

Scientific Notation

Example K. Calculate. Give the answer in both scientific notation and the standard notation.

a. (1.2 x 108) x (1.3 x 10–12) = 1.2 x 1.3 x 108 x 10 –12

= 1.56 x 108 –12

= 1.56 x 10 –4 = 0.000156

b. 6.3 x 10-2 2.1 x 10-10

= 6.32.1

x 10 – 2 – ( – 10)

Scientific Notation

Example K. Calculate. Give the answer in both scientific notation and the standard notation.

a. (1.2 x 108) x (1.3 x 10–12) = 1.2 x 1.3 x 108 x 10 –12

= 1.56 x 108 –12

= 1.56 x 10 –4 = 0.000156

b. 6.3 x 10-2 2.1 x 10-10

= 6.32.1

x 10 – 2 – ( – 10)

= 3 x 108

Scientific Notation

Example K. Calculate. Give the answer in both scientific notation and the standard notation.

a. (1.2 x 108) x (1.3 x 10–12) = 1.2 x 1.3 x 108 x 10 –12

= 1.56 x 108 –12

= 1.56 x 10 –4 = 0.000156

b. 6.3 x 10-2 2.1 x 10-10

= 6.32.1

x 10 – 2 – ( – 10)

= 3 x 108

= 300,000,000

Scientific Notation

Example L. Convert each numbers into scientific notation. Calculate the result. Give the answer in both scientific notation and the standard notation.

Scientific Notation

Example L. Convert each numbers into scientific notation. Calculate the result. Give the answer in both scientific notation and the standard notation.

240,000,000 x 0.0000025 0.00015

Scientific Notation

Example L. Convert each numbers into scientific notation. Calculate the result. Give the answer in both scientific notation and the standard notation.

240,000,000 x 0.0000025

=

0.00015 2.4 x 108

Scientific Notation

Example L. Convert each numbers into scientific notation. Calculate the result. Give the answer in both scientific notation and the standard notation.

240,000,000 x 0.0000025

=

0.00015 2.4 x 108 x 2.5 x 10–6

Scientific Notation

Example L. Convert each numbers into scientific notation. Calculate the result. Give the answer in both scientific notation and the standard notation.

240,000,000 x 0.0000025

=

0.00015 2.4 x 108 x 2.5 x 10–6

1.5 x 10–4

Scientific Notation

Example L. Convert each numbers into scientific notation. Calculate the result. Give the answer in both scientific notation and the standard notation.

240,000,000 x 0.0000025

=

0.00015 2.4 x 108 x 2.5 x 10–6

1.5 x 10–4

= 2.4 x 2.5 x 108 x 10–6

1.5 x 10–4

Scientific Notation

Example L. Convert each numbers into scientific notation. Calculate the result. Give the answer in both scientific notation and the standard notation.

240,000,000 x 0.0000025

=

0.00015 2.4 x 108 x 2.5 x 10–6

1.5 x 10–4

= 2.4 x 2.51.5

x 10 8 + (–6) – ( – 4)

= 2.4 x 2.5 x 108 x 10–6

1.5 x 10–4

Scientific Notation

Example L. Convert each numbers into scientific notation. Calculate the result. Give the answer in both scientific notation and the standard notation.

240,000,000 x 0.0000025

=

0.00015 2.4 x 108 x 2.5 x 10–6

1.5 x 10–4

= 2.4 x 2.51.5

x 10 8 + (–6) – ( – 4)

= 2.4 x 2.5 x 108 x 10–6

1.5 x 10–4

= 4 x 108 – 6 + 4

Scientific Notation

Example L. Convert each numbers into scientific notation. Calculate the result. Give the answer in both scientific notation and the standard notation.

240,000,000 x 0.0000025

=

0.00015 2.4 x 108 x 2.5 x 10–6

1.5 x 10–4

= 2.4 x 2.51.5

x 10 8 + (–6) – ( – 4)

= 2.4 x 2.5 x 108 x 10–6

1.5 x 10–4

= 4 x 108 – 6 + 4

= 4 x 106 = 4,000,000

Scientific Notation

Ex. A. Write the numbers without the negative exponents and compute the answers. 1. 2–1 2. –2–2 3. 2–3 4. (–3)–2 5. 3–3

6. 5–2 7. 4–3 8. 12( )–3

9. 23( )–1

10. 32( )–2

11. 2–1* 3–2 12. 2–2+ 3–1 13. 2 * 4–1– 50 * 3–1

14. 32 * 6–1– 6 * 2–3 15. 2–2* 3–1 + 80 * 2–1

Ex. B. Combine the exponents. Leave the answers in positive exponents–but do not reciprocate the negative exponents until the final step. 16. x3x5 17. x–3x5 18. x3x–5 19. x–3x–5

20. x4y2x3y–4 21. y–3x–2 y–4x4 22. 22x–3xy2x32–5

23. 32y–152–2x5y2x–9 24. 42x252–3y–34 x–41y–11

25. x2(x3)5 26. (x–3)–5x –6 27. x4(x3y–5) –3y–8

Exponents

x–8

x–3

B. Combine the exponents. Leave the answers in positive exponents–but do not reciprocate the negative exponents until the final step.

28. x8

x–3 29.x–8

x3 30. y6x–8

x–2y3 31.

x6x–2y–8

y–3x–5y2 32.2–3x6y–8

2–5y–5x2 33. 3–2y2x4

2–3x3y–2 34.

4–1(x3y–2)–2

2–3(y–5x2)–1 35.6–2 y2(x4y–3)–1

9–1(x3y–2)–4y236.

C. Combine the exponents as much as possible.

38. 232x 39. 3x+23x 40. ax–3ax+5

41. (b2)x+1b–x+3 42. e3e2x+1e–x

43. e3e2x+1e–x

44. How would you make sense of 23 ?2