0. exponents y
Transcript of 0. exponents y
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Exponents
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ExponentsMultiplying A to 1 repeatedly N times is written as AN.
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ExponentsMultiplying A to 1 repeatedly N times is written as AN.
N times
1 x A x A x A ….x A = AN
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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0-power Rule: A0 = 1 (A≠0)
Special Exponents
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0-power Rule: A0 = 1 (A=0)
Special Exponents
because 1 = A1
A1
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0-power Rule: A0 = 1 (A=0)
Special Exponents
because 1 = = A1–1 = A0A1
A1(divide–subtract)
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0-power Rule: A0 = 1 (A=0)
Special Exponents
because 1 = = A1–1 = A0A1
A1(divide–subtract)
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0-power Rule: A0 = 1 (A=0)
1Ak
Special Exponents
because 1 = = A1–1 = A0A1
A1
Negative Power Rule: A–k =
(divide–subtract)
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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)
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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)
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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)
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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)
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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)
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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)
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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)
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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)
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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)
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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)
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Special Exponents By the power–multiply rule, the fractional exponent
Akn
±
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Special Exponents By the power–multiply rule, the fractional exponent
Akn
±
(A )n1
is
take the nth root of A
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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
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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
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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 =
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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 =
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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 =
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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 =
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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 =
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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 =
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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
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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
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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
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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
![Page 47: 0. exponents y](https://reader031.fdocuments.us/reader031/viewer/2022022415/5a6490697f8b9a82568b4aeb/html5/thumbnails/47.jpg)
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
![Page 48: 0. exponents y](https://reader031.fdocuments.us/reader031/viewer/2022022415/5a6490697f8b9a82568b4aeb/html5/thumbnails/48.jpg)
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
![Page 49: 0. exponents y](https://reader031.fdocuments.us/reader031/viewer/2022022415/5a6490697f8b9a82568b4aeb/html5/thumbnails/49.jpg)
a.16–½ =
Fractional Powers
b. 43/2 =
Your turn: calculate the root, then raise the root to the
numerator–power.
![Page 50: 0. exponents y](https://reader031.fdocuments.us/reader031/viewer/2022022415/5a6490697f8b9a82568b4aeb/html5/thumbnails/50.jpg)
a.16–½ =
Fractional Powers
b. 43/2 =
Your turn: calculate the root, then raise the root to the
numerator–power.
Ans: ¼, 8
![Page 51: 0. exponents y](https://reader031.fdocuments.us/reader031/viewer/2022022415/5a6490697f8b9a82568b4aeb/html5/thumbnails/51.jpg)
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.
![Page 52: 0. exponents y](https://reader031.fdocuments.us/reader031/viewer/2022022415/5a6490697f8b9a82568b4aeb/html5/thumbnails/52.jpg)
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.
![Page 53: 0. exponents y](https://reader031.fdocuments.us/reader031/viewer/2022022415/5a6490697f8b9a82568b4aeb/html5/thumbnails/53.jpg)
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.
![Page 54: 0. exponents y](https://reader031.fdocuments.us/reader031/viewer/2022022415/5a6490697f8b9a82568b4aeb/html5/thumbnails/54.jpg)
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
![Page 55: 0. exponents y](https://reader031.fdocuments.us/reader031/viewer/2022022415/5a6490697f8b9a82568b4aeb/html5/thumbnails/55.jpg)
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
![Page 56: 0. exponents y](https://reader031.fdocuments.us/reader031/viewer/2022022415/5a6490697f8b9a82568b4aeb/html5/thumbnails/56.jpg)
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
![Page 57: 0. exponents y](https://reader031.fdocuments.us/reader031/viewer/2022022415/5a6490697f8b9a82568b4aeb/html5/thumbnails/57.jpg)
Fractional PowersOften it’s easier to manipulate radical–expressions
using the fractional exponent notation.
![Page 58: 0. exponents y](https://reader031.fdocuments.us/reader031/viewer/2022022415/5a6490697f8b9a82568b4aeb/html5/thumbnails/58.jpg)
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
![Page 59: 0. exponents y](https://reader031.fdocuments.us/reader031/viewer/2022022415/5a6490697f8b9a82568b4aeb/html5/thumbnails/59.jpg)
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 =
![Page 60: 0. exponents y](https://reader031.fdocuments.us/reader031/viewer/2022022415/5a6490697f8b9a82568b4aeb/html5/thumbnails/60.jpg)
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 =
![Page 61: 0. exponents y](https://reader031.fdocuments.us/reader031/viewer/2022022415/5a6490697f8b9a82568b4aeb/html5/thumbnails/61.jpg)
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
![Page 62: 0. exponents y](https://reader031.fdocuments.us/reader031/viewer/2022022415/5a6490697f8b9a82568b4aeb/html5/thumbnails/62.jpg)
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
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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
![Page 64: 0. exponents y](https://reader031.fdocuments.us/reader031/viewer/2022022415/5a6490697f8b9a82568b4aeb/html5/thumbnails/64.jpg)
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
![Page 65: 0. exponents y](https://reader031.fdocuments.us/reader031/viewer/2022022415/5a6490697f8b9a82568b4aeb/html5/thumbnails/65.jpg)
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
![Page 66: 0. exponents y](https://reader031.fdocuments.us/reader031/viewer/2022022415/5a6490697f8b9a82568b4aeb/html5/thumbnails/66.jpg)
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
![Page 67: 0. exponents y](https://reader031.fdocuments.us/reader031/viewer/2022022415/5a6490697f8b9a82568b4aeb/html5/thumbnails/67.jpg)
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
![Page 68: 0. exponents y](https://reader031.fdocuments.us/reader031/viewer/2022022415/5a6490697f8b9a82568b4aeb/html5/thumbnails/68.jpg)
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
![Page 69: 0. exponents y](https://reader031.fdocuments.us/reader031/viewer/2022022415/5a6490697f8b9a82568b4aeb/html5/thumbnails/69.jpg)
Decimal PowersWe write decimal–exponents as fractional exponents,
then we extract roots and raise powers as indicated
by the fractional exponents.
![Page 70: 0. exponents y](https://reader031.fdocuments.us/reader031/viewer/2022022415/5a6490697f8b9a82568b4aeb/html5/thumbnails/70.jpg)
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 =
![Page 71: 0. exponents y](https://reader031.fdocuments.us/reader031/viewer/2022022415/5a6490697f8b9a82568b4aeb/html5/thumbnails/71.jpg)
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 =
![Page 72: 0. exponents y](https://reader031.fdocuments.us/reader031/viewer/2022022415/5a6490697f8b9a82568b4aeb/html5/thumbnails/72.jpg)
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 =
![Page 73: 0. exponents y](https://reader031.fdocuments.us/reader031/viewer/2022022415/5a6490697f8b9a82568b4aeb/html5/thumbnails/73.jpg)
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 =
![Page 74: 0. exponents y](https://reader031.fdocuments.us/reader031/viewer/2022022415/5a6490697f8b9a82568b4aeb/html5/thumbnails/74.jpg)
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 =
![Page 75: 0. exponents y](https://reader031.fdocuments.us/reader031/viewer/2022022415/5a6490697f8b9a82568b4aeb/html5/thumbnails/75.jpg)
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 =
![Page 76: 0. exponents y](https://reader031.fdocuments.us/reader031/viewer/2022022415/5a6490697f8b9a82568b4aeb/html5/thumbnails/76.jpg)
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 =
![Page 77: 0. exponents y](https://reader031.fdocuments.us/reader031/viewer/2022022415/5a6490697f8b9a82568b4aeb/html5/thumbnails/77.jpg)
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
![Page 78: 0. exponents y](https://reader031.fdocuments.us/reader031/viewer/2022022415/5a6490697f8b9a82568b4aeb/html5/thumbnails/78.jpg)
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.
![Page 79: 0. exponents y](https://reader031.fdocuments.us/reader031/viewer/2022022415/5a6490697f8b9a82568b4aeb/html5/thumbnails/79.jpg)
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
![Page 80: 0. exponents y](https://reader031.fdocuments.us/reader031/viewer/2022022415/5a6490697f8b9a82568b4aeb/html5/thumbnails/80.jpg)
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
![Page 81: 0. exponents y](https://reader031.fdocuments.us/reader031/viewer/2022022415/5a6490697f8b9a82568b4aeb/html5/thumbnails/81.jpg)
Fractional Powers