Exponents

http://www.lahc.edu/math/precalculus/math_260a.html

ExponentsMultiplying A to 1 repeatedly N times is written as AN.

ExponentsMultiplying A to 1 repeatedly N times is written as AN.

N times

1 x A x A x A ….x A = AN

ExponentsMultiplying A to 1 repeatedly N times is written as AN.

A is the base.

N is the exponent.

N times

1 x A x A x A ….x A = AN

Multiply–Add Rule:Divide–Subtract Rule:

Power–Multiply Rule:

Exponents

Exponent–Rules

Multiplying A to 1 repeatedly N times is written as AN.

A is the base.

N is the exponent.

N times

1 x A x A x A ….x A = AN

Multiply–Add Rule: AnAk = An+k Divide–Subtract Rule:

Power–Multiply Rule:

Exponents

Exponent–Rules

Multiplying A to 1 repeatedly N times is written as AN.

A is the base.

N is the exponent.

N times

1 x A x A x A ….x A = AN

Multiply–Add Rule: AnAk = An+k Divide–Subtract Rule:

Example A. a. 5254 =

Power–Multiply Rule:

Exponents

Exponent–Rules

Multiplying A to 1 repeatedly N times is written as AN.

A is the base.

N is the exponent.

N times

1 x A x A x A ….x A = AN

Multiply–Add Rule: AnAk = An+k Divide–Subtract Rule:

Example A. a. 5254 = (5*5)(5*5*5*5)

Power–Multiply Rule:

Exponents

Exponent–Rules

Multiplying A to 1 repeatedly N times is written as AN.

A is the base.

N is the exponent.

N times

1 x A x A x A ….x A = AN

Multiply–Add Rule: AnAk = An+k Divide–Subtract Rule:

Example A. a. 5254 = (5*5)(5*5*5*5) = 52 + 4 = 56

Power–Multiply Rule:

Exponents

(multiply–add)

Exponent–Rules

Multiplying A to 1 repeatedly N times is written as AN.

A is the base.

N is the exponent.

N times

1 x A x A x A ….x A = AN

Multiply–Add Rule: AnAk = An+k Divide–Subtract Rule:

Example A. a. 5254 = (5*5)(5*5*5*5) = 52 + 4 = 56

An

Ak

Power–Multiply Rule:

Exponents

= An – k

(multiply–add)

Exponent–Rules

Multiplying A to 1 repeatedly N times is written as AN.

A is the base.

N is the exponent.

N times

1 x A x A x A ….x A = AN

Multiply–Add Rule: AnAk = An+k Divide–Subtract Rule:

Example A. a. 5254 = (5*5)(5*5*5*5) = 52 + 4 = 56

An

Ak

b. =55

52

Power–Multiply Rule:

Exponents

= An – k

(multiply–add)

Exponent–Rules

Multiplying A to 1 repeatedly N times is written as AN.

A is the base.

N is the exponent.

N times

1 x A x A x A ….x A = AN

Multiply–Add Rule: AnAk = An+k Divide–Subtract Rule:

Example A. a. 5254 = (5*5)(5*5*5*5) = 52 + 4 = 56

An

Ak

b. = 55–2 = 53 55

52

Power–Multiply Rule:

Exponents

= An – k

(multiply–add)

Exponent–Rules

Multiplying A to 1 repeatedly N times is written as AN.

A is the base.

N is the exponent.

N times

1 x A x A x A ….x A = AN

Multiply–Add Rule: AnAk = An+k Divide–Subtract Rule:

Example A. a. 5254 = (5*5)(5*5*5*5) = 52 + 4 = 56

An

Ak

b. = 55–2 = 53 55

52

Power–Multiply Rule:

Exponents

= An – k

(multiply–add)

(divide–subtract)

Exponent–Rules

Multiplying A to 1 repeatedly N times is written as AN.

A is the base.

N is the exponent.

N times

1 x A x A x A ….x A = AN

Multiply–Add Rule: AnAk = An+k Divide–Subtract Rule:

Example A. a. 5254 = (5*5)(5*5*5*5) = 52 + 4 = 56

An

Ak

b. = 55–2 = 53 55

52

Power–Multiply Rule: (An)k = Ank , (An Bm)k = Ank Bmk

Exponents

= An – k

(multiply–add)

(divide–subtract)

Exponent–Rules

Multiplying A to 1 repeatedly N times is written as AN.

A is the base.

N is the exponent.

N times

1 x A x A x A ….x A = AN

Multiply–Add Rule: AnAk = An+k Divide–Subtract Rule:

Example A. a. 5254 = (5*5)(5*5*5*5) = 52 + 4 = 56

An

Ak

b. = 55–2 = 53 55

52

Power–Multiply Rule: (An)k = Ank , (An Bm)k = Ank Bmk

c. (22*34)3 =

Exponents

= An – k

(multiply–add)

(divide–subtract)

Exponent–Rules

Multiplying A to 1 repeatedly N times is written as AN.

A is the base.

N is the exponent.

N times

1 x A x A x A ….x A = AN

Multiply–Add Rule: AnAk = An+k Divide–Subtract Rule:

Example A. a. 5254 = (5*5)(5*5*5*5) = 52 + 4 = 56

An

Ak

b. = 55–2 = 53 55

52

Power–Multiply Rule: (An)k = Ank , (An Bm)k = Ank Bmk

c. (22*34)3 = 26*312

Exponents

= An – k

(multiply–add)

(divide–subtract)(power–multiply)

Exponent–Rules

Multiplying A to 1 repeatedly N times is written as AN.

A is the base.

N is the exponent.

N times

1 x A x A x A ….x A = AN

Multiply–Add Rule: AnAk = An+k Divide–Subtract Rule:

Example A. a. 5254 = (5*5)(5*5*5*5) = 52 + 4 = 56

An

Ak

b. = 55–2 = 53 55

52

Power–Multiply Rule: (An)k = Ank , (An Bm)k = Ank Bmk

c. (22*34)3 = 26*312

Exponents

= An – k

(multiply–add)

(divide–subtract)(power–multiply)

Exponent–Rules

! Note that (22 ± 34)3 = 26 ± 38

Multiplying A to 1 repeatedly N times is written as AN.

A is the base.

N is the exponent.

N times

1 x A x A x A ….x A = AN

0-power Rule: A0 = 1 (A≠0)Special Exponents

0-power Rule: A0 = 1 (A=0)Special Exponents

because 1 = A1

A1

0-power Rule: A0 = 1 (A=0)Special Exponents

because 1 = = A1–1 = A0A1

A1 (divide–subtract)

0-power Rule: A0 = 1 (A=0)Special Exponents

because 1 = = A1–1 = A0A1

A1 (divide–subtract)

0-power Rule: A0 = 1 (A=0)

1Ak

Special Exponents

because 1 = = A1–1 = A0A1

A1

Negative Power Rule: A–k =(divide–subtract)

0-power Rule: A0 = 1 (A=0)

=

1Ak

1Ak

A0

Ak

Special Exponents

because 1 = = A1–1 = A0A1

A1

Negative Power Rule: A–k =

because

(divide–subtract)

0-power Rule: A0 = 1 (A=0)

=

1Ak

1Ak

A0

Ak

Special Exponents

because 1 = = A1–1 = A0A1

A1

Negative Power Rule: A–k =

because = A0–k = A–k

(divide–subtract)

(divide–subtract)

0-power Rule: A0 = 1 (A=0)

=

1Ak

1Ak

A0

Ak

Special Exponents

½ - Power Rule: A½ = A , the square root of A,

because 1 = = A1–1 = A0A1

A1

Negative Power Rule: A–k =

because = A0–k = A–k

(divide–subtract)

(divide–subtract)

0-power Rule: A0 = 1 (A=0)

=

1Ak

1Ak

A0

Ak

Special Exponents

½ - Power Rule: A½ = A , the square root of A,because (A½)2 = A = (A)2,

because 1 = = A1–1 = A0A1

A1

Negative Power Rule: A–k =

because = A0–k = A–k

(divide–subtract)

(divide–subtract)

0-power Rule: A0 = 1 (A=0)

=

1Ak

1Ak

A0

Ak

Special Exponents

½ - Power Rule: A½ = A , the square root of A,because (A½)2 = A = (A)2, so A½ = A

because 1 = = A1–1 = A0A1

A1

Negative Power Rule: A–k =

because = A0–k = A–k

(divide–subtract)

(divide–subtract)

0-power Rule: A0 = 1 (A=0)

=

1Ak

1Ak

A0

Ak

Special Exponents

½ - Power Rule: A½ = A , the square root of A,because (A½)2 = A = (A)2, so A½ = A

because 1 = = A1–1 = A0A1

A1

Negative Power Rule: A–k =

because = A0–k = A–k

1/n - Power Rule: A1/n = A , the nth root of A.n

(divide–subtract)

(divide–subtract)

0-power Rule: A0 = 1 (A=0)

=

1Ak

1Ak

A0

Ak

Special Exponents

½ - Power Rule: A½ = A , the square root of A,because (A½)2 = A = (A)2, so A½ = A

Example B.

because 1 = = A1–1 = A0A1

A1

Negative Power Rule: A–k =

because = A0–k = A–k

1/n - Power Rule: A1/n = A , the nth root of A.n

c. 641/3 =b. 81/3 =

a. 641/2 =

(divide–subtract)

(divide–subtract)

0-power Rule: A0 = 1 (A=0)

=

1Ak

1Ak

A0

Ak

Special Exponents

½ - Power Rule: A½ = A , the square root of A,because (A½)2 = A = (A)2, so A½ = A

Example B.

because 1 = = A1–1 = A0A1

A1

Negative Power Rule: A–k =

because = A0–k = A–k

1/n - Power Rule: A1/n = A , the nth root of A.n

c. 641/3 = b. 81/3 =

a. 641/2 = 64 = 8

(divide–subtract)

(divide–subtract)

0-power Rule: A0 = 1 (A=0)

=

1Ak

1Ak

A0

Ak

Special Exponents

½ - Power Rule: A½ = A , the square root of A,because (A½)2 = A = (A)2, so A½ = A

Example B.

because 1 = = A1–1 = A0A1

A1

Negative Power Rule: A–k =

because = A0–k = A–k

1/n - Power Rule: A1/n = A , the nth root of A.n

c. 641/3 =b. 81/3 = 8 = 23

a. 641/2 = 64 = 8

(divide–subtract)

(divide–subtract)

0-power Rule: A0 = 1 (A=0)

=

1Ak

1Ak

A0

Ak

Special Exponents

½ - Power Rule: A½ = A , the square root of A,because (A½)2 = A = (A)2, so A½ = A

Example B.

because 1 = = A1–1 = A0A1

A1

Negative Power Rule: A–k =

because = A0–k = A–k

1/n - Power Rule: A1/n = A , the nth root of A.n

c. 641/3 = 64 = 43

b. 81/3 = 8 = 23 a. 641/2 = 64 = 8

(divide–subtract)

(divide–subtract)

Special Exponents By the power–multiply rule, the fractional exponent

A kn±

Special Exponents By the power–multiply rule, the fractional exponent

A kn± (A )n

1is

take the nth root of A

Special Exponents By the power–multiply rule, the fractional exponent

A kn± (A ) kn ±1

is

take the nth root of A

then raise the root to ±k power

Special Exponents

To calculate a fractional power: extract the root first, then raise the root to the numerator–power.

By the power–multiply rule, the fractional exponent

A kn± (A ) kn ±1

is

take the nth root of A

then raise the root to ±k power

Special Exponents

a. 9 –3/2 =

To calculate a fractional power: extract the root first, then raise the root to the numerator–power.

Example C. Find the root, then raise the root to the numerator–power.

By the power–multiply rule, the fractional exponent

A kn± (A ) kn ±1

is

take the nth root of A

then raise the root to ±k power

c. 16 -3/4 =b. 27 -2/3 =

Special Exponents

a. 9 –3/2 = (9 ½ * –3)

To calculate a fractional power: extract the root first, then raise the root to the numerator–power.

Example C. Find the root, then raise the root to the numerator–power.

By the power–multiply rule, the fractional exponent

A kn± (A ) kn ±1

is

take the nth root of A

then raise the root to ±k power

c. 16 -3/4 =b. 27 -2/3 =

Special Exponents

a. 9 –3/2 = (9 ½ * –3) = (9½)–3

To calculate a fractional power: extract the root first, then raise the root to the numerator–power.

Example C. Find the root, then raise the root to the numerator–power.

By the power–multiply rule, the fractional exponent

A kn± (A ) kn ±1

is

take the nth root of A

then raise the root to ±k power

c. 16 -3/4 =b. 27 -2/3 =

Special Exponents

a. 9 –3/2 = (9 ½ * –3) = (9½)–3 = (9)–3

To calculate a fractional power: extract the root first, then raise the root to the numerator–power.

Example C. Find the root, then raise the root to the numerator–power.

By the power–multiply rule, the fractional exponent

A kn± (A ) kn ±1

is

take the nth root of A

then raise the root to ±k power

c. 16 -3/4 =b. 27 -2/3 =

Special Exponents

a. 9 –3/2 = (9 ½ * –3) = (9½)–3 = (9)–3 = 3–3

To calculate a fractional power: extract the root first, then raise the root to the numerator–power.

Example C. Find the root, then raise the root to the numerator–power.

By the power–multiply rule, the fractional exponent

A kn± (A ) kn ±1

is

take the nth root of A

then raise the root to ±k power

c. 16 -3/4 =b. 27 -2/3 =

Special Exponents

a. 9 –3/2 = (9 ½ * –3) = (9½)–3 = (9)–3 = 3–3 = 1/27

To calculate a fractional power: extract the root first, then raise the root to the numerator–power.

Example C. Find the root, then raise the root to the numerator–power.

By the power–multiply rule, the fractional exponent

A kn± (A ) kn ±1

is

take the nth root of A

then raise the root to ±k power

c. 16 -3/4 =b. 27 -2/3 =

Special Exponents

a. 9 –3/2 = (9 ½ * –3) = (9½)–3 = (9)–3 = 3–3 = 1/27

To calculate a fractional power: extract the root first, then raise the root to the numerator–power.

Example C. Find the root, then raise the root to the numerator–power.

By the power–multiply rule, the fractional exponent

A kn± (A ) kn ±1

is

take the nth root of A

then raise the root to ±k power

c. 16 -3/4 =b. 27 -2/3 = (271/3)-2 = (27)-23

Special Exponents

a. 9 –3/2 = (9 ½ * –3) = (9½)–3 = (9)–3 = 3–3 = 1/27

To calculate a fractional power: extract the root first, then raise the root to the numerator–power.

Example C. Find the root, then raise the root to the numerator–power.

By the power–multiply rule, the fractional exponent

A kn± (A ) kn ±1

is

take the nth root of A

then raise the root to ±k power

c. 16 -3/4 = b. 27 -2/3 = (271/3)-2 = (27)-2 = (3)-2 =3

Special Exponents

a. 9 –3/2 = (9 ½ * –3) = (9½)–3 = (9)–3 = 3–3 = 1/27

To calculate a fractional power: extract the root first, then raise the root to the numerator–power.

Example C. Find the root, then raise the root to the numerator–power.

By the power–multiply rule, the fractional exponent

A kn± (A ) kn ±1

is

take the nth root of A

then raise the root to ±k power

c. 16 -3/4 =b. 27 -2/3 = (271/3)-2 = (27)-2 = (3)-2 = 1/32 = 1/93

Special Exponents

a. 9 –3/2 = (9 ½ * –3) = (9½)–3 = (9)–3 = 3–3 = 1/27

To calculate a fractional power: extract the root first, then raise the root to the numerator–power.

Example C. Find the root, then raise the root to the numerator–power.

By the power–multiply rule, the fractional exponent

A kn± (A ) kn ±1

is

take the nth root of A

then raise the root to ±k power

c. 16 -3/4 = (161/4)-3 = (16)-34

b. 27 -2/3 = (271/3)-2 = (27)-2 = (3)-2 = 1/32 = 1/93

Special Exponents

a. 9 –3/2 = (9 ½ * –3) = (9½)–3 = (9)–3 = 3–3 = 1/27

To calculate a fractional power: extract the root first, then raise the root to the numerator–power.

Example C. Find the root, then raise the root to the numerator–power.

By the power–multiply rule, the fractional exponent

A kn± (A ) kn ±1

is

take the nth root of A

then raise the root to ±k power

c. 16 -3/4 = (161/4)-3 = (16)-3 = (2)-34

b. 27 -2/3 = (271/3)-2 = (27)-2 = (3)-2 = 1/32 = 1/93

Special Exponents

a. 9 –3/2 = (9 ½ * –3) = (9½)–3 = (9)–3 = 3–3 = 1/27

To calculate a fractional power: extract the root first, then raise the root to the numerator–power.

Example C. Find the root, then raise the root to the numerator–power.

By the power–multiply rule, the fractional exponent

A kn± (A ) kn ±1

is

take the nth root of A

then raise the root to ±k power

c. 16 -3/4 = (161/4)-3 = (16)-3 = (2)-3 = 1/23 = 1/84

b. 27 -2/3 = (271/3)-2 = (27)-2 = (3)-2 = 1/32 = 1/93

a.16–½ =

Fractional Powers

b. 43/2 =

Your turn: calculate the root, then raise the root to the numerator–power.

a.16–½ =

Fractional Powers

b. 43/2 =

Your turn: calculate the root, then raise the root to the numerator–power.

Ans: ¼, 8

a.16–½ =

Fractional Powers

b. 43/2 =

Your turn: calculate the root, then raise the root to the numerator–power.

Ans: ¼, 8We use the multiply–add, divide–subtract, and power–multiply rules to collect fractional exponents.

a.16–½ =

Fractional Powers

b. 43/2 =

Your turn: calculate the root, then raise the root to the numerator–power.

Ans: ¼, 8We use the multiply–add, divide–subtract, and power–multiply rules to collect fractional exponents.

x*(x1/3y3/2)2

x–1/2y2/3 =

Example D. Simplify by combining the exponents.

a.16–½ =

Fractional Powers

b. 43/2 =

Your turn: calculate the root, then raise the root to the numerator–power.

Ans: ¼, 8We use the multiply–add, divide–subtract, and power–multiply rules to collect fractional exponents.

x*(x1/3y3/2)2

x–1/2y2/3 =x*x2/3y3

x–1/2y2/3

power–multiply rule1/3*2 3/2*2

Example D. Simplify by combining the exponents.

a.16–½ =

Fractional Powers

b. 43/2 =

Your turn: calculate the root, then raise the root to the numerator–power.

Ans: ¼, 8We use the multiply–add, divide–subtract, and power–multiply rules to collect fractional exponents.

x*(x1/3y3/2)2

x–1/2y2/3 =x*x2/3y3

x–1/2y2/3 = x–1/2y2/3x5/3y3

Example D. Simplify by combining the exponents.power–multiply rule

1/3*2 3/2*2multiply–add rule

1 + 2/3

a.16–½ =

Fractional Powers

b. 43/2 =

Your turn: calculate the root, then raise the root to the numerator–power.

Ans: ¼, 8We use the multiply–add, divide–subtract, and power–multiply rules to collect fractional exponents.

x*(x1/3y3/2)2

x–1/2y2/3 =x*x2/3y3

x–1/2y2/3 = x–1/2y2/3

=

x5/3y3

x5/3 – (–1/2) y3 – 2/3

Example D. Simplify by combining the exponents.power–multiply rule

1/3*2 3/2*2multiply–add rule

1 + 2/3

divide–subtract rule

a.16–½ =

Fractional Powers

b. 43/2 =

Your turn: calculate the root, then raise the root to the numerator–power.

Ans: ¼, 8We use the multiply–add, divide–subtract, and power–multiply rules to collect fractional exponents.

x*(x1/3y3/2)2

x–1/2y2/3 =x*x2/3y3

x–1/2y2/3 = x–1/2y2/3

=

x5/3y3

x5/3 – (–1/2) y3 – 2/3

= x13/6 y7/3

Example D. Simplify by combining the exponents.power–multiply rule

1/3*2 3/2*2multiply–add rule

1 + 2/3

divide–subtract rule

Fractional PowersOften it’s easier to manipulate radical–expressions using the fractional exponent notation.

Fractional PowersOften it’s easier to manipulate radical–expressions using the fractional exponent notation. To write a radical in fractional exponent, assuming a is defined, we have that:k

an = ( a )n → ak k kn

Fractional PowersOften it’s easier to manipulate radical–expressions using the fractional exponent notation. To write a radical in fractional exponent, assuming a is defined, we have that:k

an = ( a )n → ak k kn

Example E. Write the following expressions using fractional exponents then simplify if possible.

c. 9 + a2 =

a. 53 or (5 )3 =b. 9a2 =

Fractional PowersOften it’s easier to manipulate radical–expressions using the fractional exponent notation. To write a radical in fractional exponent, assuming a is defined, we have that:k

an = ( a )n → ak k kn

Example E. Write the following expressions using fractional exponents then simplify if possible.

c. 9 + a2 =

a. 53 or (5 )3 = 53/2

b. 9a2 =

Fractional PowersOften it’s easier to manipulate radical–expressions using the fractional exponent notation. To write a radical in fractional exponent, assuming a is defined, we have that:k

an = ( a )n → ak k kn

Example E. Write the following expressions using fractional exponents then simplify if possible.

c. 9 + a2 =

a. 53 or (5 )3 = 53/2

b. 9a2 = (9a2)1/2

Fractional PowersOften it’s easier to manipulate radical–expressions using the fractional exponent notation. To write a radical in fractional exponent, assuming a is defined, we have that:k

an = ( a )n → ak k kn

Example E. Write the following expressions using fractional exponents then simplify if possible.

c. 9 + a2 =

a. 53 or (5 )3 = 53/2

b. 9a2 = (9a2)1/2 = 3a

Fractional PowersOften it’s easier to manipulate radical–expressions using the fractional exponent notation. To write a radical in fractional exponent, assuming a is defined, we have that:k

an = ( a )n → ak k kn

Example E. Write the following expressions using fractional exponents then simplify if possible.

c. 9 + a2 = (9 + a2)1/2

a. 53 or (5 )3 = 53/2

b. 9a2 = (9a2)1/2 = 3a

Fractional PowersOften it’s easier to manipulate radical–expressions using the fractional exponent notation. To write a radical in fractional exponent, assuming a is defined, we have that:k

an = ( a )n → ak k kn

Example E. Write the following expressions using fractional exponents then simplify if possible.

c. 9 + a2 = (9 + a2)1/2 ≠ (3 + a ).

a. 53 or (5 )3 = 53/2

b. 9a2 = (9a2)1/2 = 3a

Fractional PowersOften it’s easier to manipulate radical–expressions using the fractional exponent notation.

d. Express a2 (a ) as one radical. 3 4

To write a radical in fractional exponent, assuming a is defined, we have that:k

an = ( a )n → ak k kn

Example E. Write the following expressions using fractional exponents then simplify if possible.

c. 9 + a2 = (9 + a2)1/2 ≠ (3 + a ).

a. 53 or (5 )3 = 53/2

b. 9a2 = (9a2)1/2 = 3a

Fractional PowersOften it’s easier to manipulate radical–expressions using the fractional exponent notation.

a2 a = a2/3a1/4 3 4

To write a radical in fractional exponent, assuming a is defined, we have that:k

an = ( a )n → ak k kn

Example E. Write the following expressions using fractional exponents then simplify if possible.

c. 9 + a2 = (9 + a2)1/2 ≠ (3 + a ).

a. 53 or (5 )3 = 53/2

b. 9a2 = (9a2)1/2 = 3a

d. Express a2 (a ) as one radical. 3 4

Fractional PowersOften it’s easier to manipulate radical–expressions using the fractional exponent notation.

a2 a = a2/3a1/4 = a11/123 4

To write a radical in fractional exponent, assuming a is defined, we have that:k

an = ( a )n → ak k kn

Example E. Write the following expressions using fractional exponents then simplify if possible.

c. 9 + a2 = (9 + a2)1/2 ≠ (3 + a ).

a. 53 or (5 )3 = 53/2

b. 9a2 = (9a2)1/2 = 3a

d. Express a2 (a ) as one radical. 3 4

Fractional PowersOften it’s easier to manipulate radical–expressions using the fractional exponent notation.

a2 a = a2/3a1/4 = a11/12 = a113 4 12

To write a radical in fractional exponent, assuming a is defined, we have that:k

an = ( a )n → ak k kn

Example E. Write the following expressions using fractional exponents then simplify if possible.

c. 9 + a2 = (9 + a2)1/2 ≠ (3 + a ).

a. 53 or (5 )3 = 53/2

b. 9a2 = (9a2)1/2 = 3a

d. Express a2 (a ) as one radical. 3 4

Decimal PowersWe write decimal–exponents as fractional exponents, then we extract roots and raise powers as indicated by the fractional exponents.

Decimal PowersWe write decimal–exponents as fractional exponents, then we extract roots and raise powers as indicated by the fractional exponents.Example F. Write the following decimal exponents as fractional exponents then simplify, if possible. a. 9–1.5 =b. 16–0.75 =c. 30.4 =

Decimal PowersWe write decimal–exponents as fractional exponents, then we extract roots and raise powers as indicated by the fractional exponents.Example F. Write the following decimal exponents as fractional exponents then simplify, if possible. a. 9–1.5 = 9 –3/2

b. 16–0.75 =c. 30.4 =

Decimal PowersWe write decimal–exponents as fractional exponents, then we extract roots and raise powers as indicated by the fractional exponents.Example F. Write the following decimal exponents as fractional exponents then simplify, if possible. a. 9–1.5 = 9 –3/2 = (9)–3

b. 16–0.75 =c. 30.4 =

Decimal PowersWe write decimal–exponents as fractional exponents, then we extract roots and raise powers as indicated by the fractional exponents.Example F. Write the following decimal exponents as fractional exponents then simplify, if possible. a. 9–1.5 = 9 –3/2 = (9)–3 = 3–3 = 1/27b. 16–0.75 =c. 30.4 =

Decimal PowersWe write decimal–exponents as fractional exponents, then we extract roots and raise powers as indicated by the fractional exponents.Example F. Write the following decimal exponents as fractional exponents then simplify, if possible. a. 9–1.5 = 9 –3/2 = (9)–3 = 3–3 = 1/27b. 16–0.75 = 16 –3/4

c. 30.4 =

Decimal PowersWe write decimal–exponents as fractional exponents, then we extract roots and raise powers as indicated by the fractional exponents.Example F. Write the following decimal exponents as fractional exponents then simplify, if possible. a. 9–1.5 = 9 –3/2 = (9)–3 = 3–3 = 1/27b. 16–0.75 = 16 –3/4 = (16)–3

4

c. 30.4 =

Decimal PowersWe write decimal–exponents as fractional exponents, then we extract roots and raise powers as indicated by the fractional exponents.Example F. Write the following decimal exponents as fractional exponents then simplify, if possible. a. 9–1.5 = 9 –3/2 = (9)–3 = 3–3 = 1/27b. 16–0.75 = 16 –3/4 = (16)–3 = (2)–3 = 1/8

4

c. 30.4 =

Decimal PowersWe write decimal–exponents as fractional exponents, then we extract roots and raise powers as indicated by the fractional exponents.Example F. Write the following decimal exponents as fractional exponents then simplify, if possible. a. 9–1.5 = 9 –3/2 = (9)–3 = 3–3 = 1/27b. 16–0.75 = 16 –3/4 = (16)–3 = (2)–3 = 1/8

4

c. 30.4 = 32/5 = (4)2 ≈ 1.55 (by calculator)5

Decimal PowersWe write decimal–exponents as fractional exponents, then we extract roots and raise powers as indicated by the fractional exponents.Example F. Write the following decimal exponents as fractional exponents then simplify, if possible. a. 9–1.5 = 9 –3/2 = (9)–3 = 3–3 = 1/27b. 16–0.75 = 16 –3/4 = (16)–3 = (2)–3 = 1/8

4

c. 30.4 = 32/5 = (4)2 ≈ 1.55 (by calculator)5

Working with real numbers and interpreting decimal exponents as fractions causes problems if the base is negative.

Decimal PowersWe write decimal–exponents as fractional exponents, then we extract roots and raise powers as indicated by the fractional exponents.Example F. Write the following decimal exponents as fractional exponents then simplify, if possible. a. 9–1.5 = 9 –3/2 = (9)–3 = 3–3 = 1/27b. 16–0.75 = 16 –3/4 = (16)–3 = (2)–3 = 1/8

4

c. 30.4 = 32/5 = (4)2 ≈ 1.55 (by calculator)5

Working with real numbers and interpreting decimal exponents as fractions causes problems if the base is negative. For example, (–32)0.2 can be viewed as (–32)1/5 = –32 = –2, or as (–32)2/10 = (–32)2 which is not defined.

5 10

Decimal PowersWe write decimal–exponents as fractional exponents, then we extract roots and raise powers as indicated by the fractional exponents.Example F. Write the following decimal exponents as fractional exponents then simplify, if possible. a. 9–1.5 = 9 –3/2 = (9)–3 = 3–3 = 1/27b. 16–0.75 = 16 –3/4 = (16)–3 = (2)–3 = 1/8

4

c. 30.4 = 32/5 = (4)2 ≈ 1.55 (by calculator)5

Working with real numbers and interpreting decimal exponents as fractions causes problems if the base is negative. For example, (–32)0.2 can be viewed as (–32)1/5 = –32 = –2, or as (–32)2/10 = (–32)2 which is not defined. To avoid this confusion, we assume the base is positive whenever a decimal exponent is used.

5 10