Embed Size (px)
Citation preview
Exponents
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 = 5355
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 = 5355
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 = 5355
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 = 5355
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 = 5355
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 = 5355
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
Akn
±
Special Exponents By the power–multiply rule, the fractional exponent
Akn
±
(A )n1
is
take the nth root of A
Special Exponents By the power–multiply rule, the fractional exponent
Akn
±
(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
Akn
±
(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
Akn
±
(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
Akn
±
(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
Akn
±
(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
Akn
±
(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
Akn
±
(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
Akn
±
(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
Akn
±
(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
Akn
±
(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
Akn
±
(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
Akn
±
(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
Akn
±
(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
Akn
±
(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, 8
We 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, 8
We 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, 8
We 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 rule
1/3*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, 8
We 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
Example D. Simplify by combining the exponents.power–multiply rule
1/3*2 3/2*2
multiply–add rule1 + 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, 8
We 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*2
multiply–add rule1 + 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, 8
We 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*2
multiply–add rule1 + 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 form,
assuming a is defined, we have that:k
an = ( a )n → ak k k
n
Fractional PowersOften it’s easier to manipulate radical–expressions
using the fractional exponent notation.
To write a radical in fractional exponent form,
assuming a is defined, we have that:k
an = ( a )n → ak k k
n
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 form,
assuming a is defined, we have that:k
an = ( a )n → ak k k
n
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 form,
assuming a is defined, we have that:k
an = ( a )n → ak k k
n
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 form,
assuming a is defined, we have that:k
an = ( a )n → ak k k
n
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 form,
assuming a is defined, we have that:k
an = ( a )n → ak k k
n
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 form,
assuming a is defined, we have that:k
an = ( a )n → ak k k
n
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 form,
assuming a is defined, we have that:k
an = ( a )n → ak k k
n
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 form,
assuming a is defined, we have that:k
an = ( a )n → ak k k
n
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 form,
assuming a is defined, we have that:k
an = ( a )n → ak k k
n
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 form,
assuming a is defined, we have that:k
an = ( a )n → ak k k
n
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 = 1/27
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 = 1/27
b. 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 = 1/27
b. 16–0.75 = 16 –3/4 = (16)–34
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 = 1/27
b. 16–0.75 = 16 –3/4 = (16)–3 = (2)–3 = 1/84
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 = 1/27
b. 16–0.75 = 16 –3/4 = (16)–3 = (2)–3 = 1/84
c. 30.4 = 32/5 = (3)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 = 1/27
b. 16–0.75 = 16 –3/4 = (16)–3 = (2)–3 = 1/84
c. 30.4 = 32/5 = (3)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 = 1/27
b. 16–0.75 = 16 –3/4 = (16)–3 = (2)–3 = 1/84
c. 30.4 = 32/5 = (3)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 = 1/27
b. 16–0.75 = 16 –3/4 = (16)–3 = (2)–3 = 1/84
c. 30.4 = 32/5 = (3)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
Fractional Powers
