Exponents
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Exponents
ExponentsWe write “1” times the quantity “A” repeatedly N times as AN
Exponents
base
exponentWe write “1” times the quantity “A” repeatedly N times as AN, i.e.
1 x A x A x A ….x A = ANrepeated N times
Example A.43
Exponents
base
exponentWe write “1” times the quantity “A” repeatedly N times as AN, i.e.
1 x A x A x A ….x A = ANrepeated N times
Example A.43 = (4)(4)(4) = 64
Exponents
base
exponentWe write “1” times the quantity “A” repeatedly N times as AN, i.e.
1 x A x A x A ….x A = ANrepeated N times
Example A.43 = (4)(4)(4) = 64 (xy)2
Exponents
base
exponentWe write “1” times the quantity “A” repeatedly N times as AN, i.e.
1 x A x A x A ….x A = ANrepeated N times
Example A.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy)
Exponents
base
exponentWe write “1” times the quantity “A” repeatedly N times as AN, i.e.
1 x A x A x A ….x A = ANrepeated N times
Example A.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2
Exponents
base
exponentWe write “1” times the quantity “A” repeatedly N times as AN, i.e.
1 x A x A x A ….x A = ANrepeated N times
Example A.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2
xy2
Exponents
base
exponentWe write “1” times the quantity “A” repeatedly N times as AN, i.e.
1 x A x A x A ….x A = ANrepeated N times
Example A.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy)
Exponents
base
exponentWe write “1” times the quantity “A” repeatedly N times as AN, i.e.
1 x A x A x A ….x A = ANrepeated N times
Example A.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy) –x2 = –(xx)
Exponents
base
exponentWe write “1” times the quantity “A” repeatedly N times as AN, i.e.
1 x A x A x A ….x A = ANrepeated N times
Example A.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy) –x2 = –(xx)
Exponents
base
exponentWe write “1” times the quantity “A” repeatedly N times as AN, i.e.
1 x A x A x A ….x A = ANrepeated N times
Example A.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy) –x2 = –(xx)
Exponents
Rules of Exponents
base
exponentWe write “1” times the quantity “A” repeatedly N times as AN, i.e.
1 x A x A x A ….x A = ANrepeated N times
Example A.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy) –x2 = –(xx)
Exponents
Multiply-Add Rule: ANAK =AN+K Rules of Exponents
base
exponentWe write “1” times the quantity “A” repeatedly N times as AN, i.e.
1 x A x A x A ….x A = ANrepeated N times
Example A.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy) –x2 = –(xx)
Exponents
Multiply-Add Rule: ANAK =AN+K Example B.
a. 5354
Rules of Exponents
base
exponentWe write “1” times the quantity “A” repeatedly N times as AN, i.e.
1 x A x A x A ….x A = ANrepeated N times
Example A.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy) –x2 = –(xx)
Exponents
Multiply-Add Rule: ANAK =AN+K Example B.
a. 5354 = (5*5*5)(5*5*5*5)
Rules of Exponents
base
exponentWe write “1” times the quantity “A” repeatedly N times as AN, i.e.
1 x A x A x A ….x A = ANrepeated N times
Example A.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy) –x2 = –(xx)
Exponents
Multiply-Add Rule: ANAK =AN+K Example B.
a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57
b. x5y7x4y6
Rules of Exponents
base
exponentWe write “1” times the quantity “A” repeatedly N times as AN, i.e.
1 x A x A x A ….x A = ANrepeated N times
Example A.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy) –x2 = –(xx)
Exponents
Multiply-Add Rule: ANAK =AN+K Example B.
a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57
b. x5y7x4y6 = x5x4y7y6
Rules of Exponents
base
exponentWe write “1” times the quantity “A” repeatedly N times as AN, i.e.
1 x A x A x A ….x A = ANrepeated N times
Example A.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy) –x2 = –(xx)
Exponents
Multiply-Add Rule: ANAK =AN+K Example B.
a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57
b. x5y7x4y6 = x5x4y7y6 = x9y13
Rules of Exponents
base
exponentWe write “1” times the quantity “A” repeatedly N times as AN, i.e.
1 x A x A x A ….x A = ANrepeated N times
Example A.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy) –x2 = –(xx)
base
exponent
Exponents
Multiply-Add Rule: ANAK =AN+K Example B.
a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57
b. x5y7x4y6 = x5x4y7y6 = x9y13
Rules of Exponents
Divide–Subtract Rule: AN
AK = AN – K
We write “1” times the quantity “A” repeatedly N times as AN, i.e.
1 x A x A x A ….x A = ANrepeated N times
Example A.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy) –x2 = –(xx)
base
exponent
Exponents
Multiply-Add Rule: ANAK =AN+K Example B.
a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57
b. x5y7x4y6 = x5x4y7y6 = x9y13
Rules of Exponents
Divide–Subtract Rule:
Example C.
AN
AK = AN – K
56
52
We write “1” times the quantity “A” repeatedly N times as AN, i.e.
1 x A x A x A ….x A = ANrepeated N times
Example A.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy) –x2 = –(xx)
base
exponent
Exponents
Multiply-Add Rule: ANAK =AN+K Example B.
a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57
b. x5y7x4y6 = x5x4y7y6 = x9y13
Rules of Exponents
Divide–Subtract Rule:
Example C.
AN
AK = AN – K
56
52 = (5)(5)(5)(5)(5)(5)(5)(5)
We write “1” times the quantity “A” repeatedly N times as AN, i.e.
1 x A x A x A ….x A = ANrepeated N times
Example A.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy) –x2 = –(xx)
base
exponent
Exponents
Multiply-Add Rule: ANAK =AN+K Example B.
a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57
b. x5y7x4y6 = x5x4y7y6 = x9y13
Rules of Exponents
Divide–Subtract Rule:
Example C.
AN
AK = AN – K
56
52 = (5)(5)(5)(5)(5)(5)(5)(5)
We write “1” times the quantity “A” repeatedly N times as AN, i.e.
1 x A x A x A ….x A = ANrepeated N times
Example A.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy) –x2 = –(xx)
base
exponent
Exponents
Multiply-Add Rule: ANAK =AN+K Example B.
a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57
b. x5y7x4y6 = x5x4y7y6 = x9y13
Rules of Exponents
Divide–Subtract Rule:
Example C.
AN
AK = AN – K
56
52 = (5)(5)(5)(5)(5)(5)(5)(5) = 56 – 2 = 54
We write “1” times the quantity “A” repeatedly N times as AN, i.e.
1 x A x A x A ….x A = ANrepeated N times
Since = 1A1
A1
Fractional Exponents
Since = 1 = A1 – 1 = A0A1
A1
Fractional Exponents
Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A1
A1
Fractional Exponents
Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A1
A1
0-Power Rule: A0 = 1
Fractional Exponents
Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A1
A1
0-Power Rule: A0 = 1
Since =1AK
A0
AK
Fractional Exponents
Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A1
A1
0-Power Rule: A0 = 1
Since = = A0 – K = A–K1AK
A0
AK
Fractional 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
Fractional 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
Fractional 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
Fractional Exponents
The reverse of multiplication is division, so 1 x A x A x …. x A = AK
A 1 x 1 x
A 1 x .. x
A 1 = A–K
repeated K times
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 D. Simplify. a. 30 = 1
Fractional Exponents
The reverse of multiplication is division, so 1 x A x A x …. x A = AK
A 1 x 1 x
A 1 x .. x
A 1 = A–K
repeated K times
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 D. Simplify.
b. 3–2
a. 30 = 1
Fractional Exponents
The reverse of multiplication is division, so 1 x A x A x …. x A = AK
A 1 x 1 x
A 1 x .. x
A 1 = A–K
repeated K times
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 D. Simplify.
1 32 b. 3–2 = =
a. 30 = 1
Fractional Exponents
The reverse of multiplication is division, so 1 x A x A x …. x A = AK
A 1 x 1 x
A 1 x .. x
A 1 = A–K
repeated K times
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 D. Simplify.
1 32
1 9b. 3–2 = =
a. 30 = 1
Fractional Exponents
The reverse of multiplication is division, so 1 x A x A x …. x A = AK
A 1 x 1 x
A 1 x .. x
A 1 = A–K
repeated K times
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 D. Simplify.
1 32
1 9
c. ( )–1 2 5
b. 3–2 = =a. 30 = 1
Fractional Exponents
The reverse of multiplication is division, so 1 x A x A x …. x A = AK
A 1 x 1 x
A 1 x .. x
A 1 = A–K
repeated K times
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 D. Simplify.
1 32
1 9
c. ( )–1 2 5 = 1
2/5
b. 3–2 = =a. 30 = 1
Fractional Exponents
The reverse of multiplication is division, so 1 x A x A x …. x A = AK
A 1 x 1 x
A 1 x .. x
A 1 = A–K
repeated K times
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 D. Simplify.
1 32
1 9
c. ( )–1 2 5 = 1
2/5 = 1* 5 2 = 5
2
b. 3–2 = =a. 30 = 1
Fractional Exponents
The reverse of multiplication is division, so 1 x A x A x …. x A = AK
A 1 x 1 x
A 1 x .. x
A 1 = A–K
repeated K times
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 D. Simplify.
1 32
1 9
c. ( )–1 2 5 = 1
2/5 = 1* 5 2 = 5
2
b. 3–2 = =a. 30 = 1
Fractional Powers
Fractional Exponents
The reverse of multiplication is division, so 1 x A x A x …. x A = AK
A 1 x 1 x
A 1 x .. x
A 1 = A–K
repeated K times
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 D. Simplify.
1 32
1 9
c. ( )–1 2 5 = 1
2/5 = 1* 5 2 = 5
2
b. 3–2 = =a. 30 = 1
Fractional Powers
Fractional Exponents
The reverse of multiplication is division, so 1 x A x A x …. x A = AK
A 1 x 1 x
A 1 x .. x
A 1 = A–K
repeated K times
1/2-Power Rule: A½ = A.
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 D. Simplify.
1 32
1 9
c. ( )–1 2 5 = 1
2/5 = 1* 5 2 = 5
2
b. 3–2 = =a. 30 = 1
Fractional Powers
Fractional Exponents
The reverse of multiplication is division, so 1 x A x A x …. x A = AK
A 1 x 1 x
A 1 x .. x
A 1 = A–K
repeated K times
By the power-multiply rule (A½ )2 = A1
1/2-Power Rule: A½ = A.
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 D. Simplify.
1 32
1 9
c. ( )–1 2 5 = 1
2/5 = 1* 5 2 = 5
2
b. 3–2 = =a. 30 = 1
Fractional Powers
Fractional Exponents
The reverse of multiplication is division, so 1 x A x A x …. x A = AK
A 1 x 1 x
A 1 x .. x
A 1 = A–K
repeated K times
By the power-multiply rule (A½ )2 = A1 = (A)2, 1/2-Power Rule: A½ = A.
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 D. Simplify.
1 32
1 9
c. ( )–1 2 5 = 1
2/5 = 1* 5 2 = 5
2
b. 3–2 = =a. 30 = 1
Fractional Powers
Fractional Exponents
The reverse of multiplication is division, so 1 x A x A x …. x A = AK
A 1 x 1 x
A 1 x .. x
A 1 = A–K
repeated K times
By the power-multiply rule (A½ )2 = A1 = (A)2, therefore A½ = A.
1/2-Power Rule: A½ = A.
Example E.a. 16½
Fractional Exponents
Example E.a. 16½ = 16
Fractional Exponents
Example E.a. 16½ = 16 = 4
Fractional Exponents
Example E.a. 16½ = 16 = 4b. 9½ 25½
Fractional Exponents
Example E.a. 16½ = 16 = 4b. 9½ 25½ =925
Fractional Exponents
Example E.a. 16½ = 16 = 4b. 9½ 25½ =925 = 3*5 = 15
Fractional Exponents
Example E.a. 16½ = 16 = 4b. 9½ 25½ =925 = 3*5 = 15c. 43/2
Fractional Exponents
Example E.a. 16½ = 16 = 4b. 9½ 25½ =925 = 3*5 = 15c. 43/2 = (4½ * 3)
Fractional Exponents
Example E.a. 16½ = 16 = 4b. 9½ 25½ =925 = 3*5 = 15c. 43/2 = (4½ * 3) = (4½)3
Fractional Exponents
Example E.a. 16½ = 16 = 4b. 9½ 25½ =925 = 3*5 = 15c. 43/2 = (4½ * 3) = (4½)3 = (4)3
Fractional Exponents
Example E.a. 16½ = 16 = 4b. 9½ 25½ =925 = 3*5 = 15c. 43/2 = (4½ * 3) = (4½)3 = (4)3 = 23 = 8
Fractional Exponents
Example E.a. 16½ = 16 = 4b. 9½ 25½ =925 = 3*5 = 15c. 43/2 = (4½ * 3) = (4½)3 = (4)3 = 23 = 8
d. 16 – ½
Fractional Exponents
Example E.a. 16½ = 16 = 4b. 9½ 25½ =925 = 3*5 = 15c. 43/2 = (4½ * 3) = (4½)3 = (4)3 = 23 = 8
d. 16 – ½ = (16½) –1
Fractional Exponents
Example E.a. 16½ = 16 = 4b. 9½ 25½ =925 = 3*5 = 15c. 43/2 = (4½ * 3) = (4½)3 = (4)3 = 23 = 8
d. 16 – ½ = (16½) –1 = (16) –1
Fractional Exponents
Example E.a. 16½ = 16 = 4b. 9½ 25½ =925 = 3*5 = 15c. 43/2 = (4½ * 3) = (4½)3 = (4)3 = 23 = 8
d. 16 – ½ = (16½) –1 = (16) –1 = 4 –1 = 1/4
Fractional Exponents
Example E.a. 16½ = 16 = 4b. 9½ 25½ =925 = 3*5 = 15c. 43/2 = (4½ * 3) = (4½)3 = (4)3 = 23 = 8
d. 16 – ½ = (16½) –1 = (16) –1 = 4 –1 = 1/4e. 9 –3/2
Fractional Exponents
Example E.a. 16½ = 16 = 4b. 9½ 25½ =925 = 3*5 = 15c. 43/2 = (4½ * 3) = (4½)3 = (4)3 = 23 = 8
d. 16 – ½ = (16½) –1 = (16) –1 = 4 –1 = 1/4e. 9 –3/2 = (9 –½ * 3)
Fractional Exponents
Example E.a. 16½ = 16 = 4b. 9½ 25½ =925 = 3*5 = 15c. 43/2 = (4½ * 3) = (4½)3 = (4)3 = 23 = 8
d. 16 – ½ = (16½) –1 = (16) –1 = 4 –1 = 1/4e. 9 –3/2 = (9 –½ * 3) = (9 ½) –3
Fractional Exponents
Example E.a. 16½ = 16 = 4b. 9½ 25½ =925 = 3*5 = 15c. 43/2 = (4½ * 3) = (4½)3 = (4)3 = 23 = 8
d. 16 – ½ = (16½) –1 = (16) –1 = 4 –1 = 1/4e. 9 –3/2 = (9 –½ * 3) = (9 ½) –3 = (9)–3
Fractional Exponents
Example E.a. 16½ = 16 = 4b. 9½ 25½ =925 = 3*5 = 15c. 43/2 = (4½ * 3) = (4½)3 = (4)3 = 23 = 8
d. 16 – ½ = (16½) –1 = (16) –1 = 4 –1 = 1/4e. 9 –3/2 = (9 –½ * 3) = (9 ½) –3 = (9)–3 = 3–3 = 1/27
Fractional Exponents
Example E.a. 16½ = 16 = 4b. 9½ 25½ =925 = 3*5 = 15c. 43/2 = (4½ * 3) = (4½)3 = (4)3 = 23 = 8
d. 16 – ½ = (16½) –1 = (16) –1 = 4 –1 = 1/4e. 9 –3/2 = (9 –½ * 3) = (9 ½) –3 = (9)–3 = 3–3 = 1/27
A–1/2 =A1In general,
Fractional Exponents
Example E.a. 16½ = 16 = 4b. 9½ 25½ =925 = 3*5 = 15c. 43/2 = (4½ * 3) = (4½)3 = (4)3 = 23 = 8
d. 16 – ½ = (16½) –1 = (16) –1 = 4 –1 = 1/4e. 9 –3/2 = (9 –½ * 3) = (9 ½) –3 = (9)–3 = 3–3 = 1/27
A–1/2 = andA1 A –k/2 = ( )k
A1In general,
Fractional Exponents
Example E.a. 16½ = 16 = 4b. 9½ 25½ =925 = 3*5 = 15c. 43/2 = (4½ * 3) = (4½)3 = (4)3 = 23 = 8
d. 16 – ½ = (16½) –1 = (16) –1 = 4 –1 = 1/4e. 9 –3/2 = (9 –½ * 3) = (9 ½) –3 = (9)–3 = 3–3 = 1/27
A–1/2 = andA1 A –k/2 = ( )k
A1
Using an argument similar to the argument for ½-power rule, we have the following rule for 1/k power.
In general,
Fractional Exponents
Example E.a. 16½ = 16 = 4b. 9½ 25½ =925 = 3*5 = 15c. 43/2 = (4½ * 3) = (4½)3 = (4)3 = 23 = 8
d. 16 – ½ = (16½) –1 = (16) –1 = 4 –1 = 1/4e. 9 –3/2 = (9 –½ * 3) = (9 ½) –3 = (9)–3 = 3–3 = 1/27
A–1/2 = andA1
1/k-Power Rule A1/k = Ak
A –k/2 = ( )k
A1
Using an argument similar to the argument for ½-power rule, we have the following rule for 1/k power.
In general,
Fractional Exponents
Example E.a. 16½ = 16 = 4b. 9½ 25½ =925 = 3*5 = 15c. 43/2 = (4½ * 3) = (4½)3 = (4)3 = 23 = 8
d. 16 – ½ = (16½) –1 = (16) –1 = 4 –1 = 1/4e. 9 –3/2 = (9 –½ * 3) = (9 ½) –3 = (9)–3 = 3–3 = 1/27
A–1/2 = andA1
1/k-Power Rule A1/k = Ak
A –k/2 = ( )k
A1
Using an argument similar to the argument for ½-power rule, we have the following rule for 1/k power.
In general,
Example F. Simplify.a. 81/3
Fractional Exponents
Example E.a. 16½ = 16 = 4b. 9½ 25½ =925 = 3*5 = 15c. 43/2 = (4½ * 3) = (4½)3 = (4)3 = 23 = 8
d. 16 – ½ = (16½) –1 = (16) –1 = 4 –1 = 1/4e. 9 –3/2 = (9 –½ * 3) = (9 ½) –3 = (9)–3 = 3–3 = 1/27
A–1/2 = andA1
1/k-Power Rule A1/k = Ak
A –k/2 = ( )k
A1
Using an argument similar to the argument for ½-power rule, we have the following rule for 1/k power.
In general,
Example F. Simplify.a. 81/3 = 83
Fractional Exponents
Example E.a. 16½ = 16 = 4b. 9½ 25½ =925 = 3*5 = 15c. 43/2 = (4½ * 3) = (4½)3 = (4)3 = 23 = 8
d. 16 – ½ = (16½) –1 = (16) –1 = 4 –1 = 1/4e. 9 –3/2 = (9 –½ * 3) = (9 ½) –3 = (9)–3 = 3–3 = 1/27
A–1/2 = andA1
1/k-Power Rule A1/k = Ak
A –k/2 = ( )k
A1
Using an argument similar to the argument for ½-power rule, we have the following rule for 1/k power.
In general,
Example F. Simplify.a. 81/3 = 8 = 23
Fractional Exponents
Example E.a. 16½ = 16 = 4b. 9½ 25½ =925 = 3*5 = 15c. 43/2 = (4½ * 3) = (4½)3 = (4)3 = 23 = 8
d. 16 – ½ = (16½) –1 = (16) –1 = 4 –1 = 1/4e. 9 –3/2 = (9 –½ * 3) = (9 ½) –3 = (9)–3 = 3–3 = 1/27
A–1/2 = andA1
1/k-Power Rule A1/k = Ak
A –k/2 = ( )k
A1
Using an argument similar to the argument for ½-power rule, we have the following rule for 1/k power.
In general,
Example F. Simplify.a. 81/3 = 8 = 2b. 27 –2/3
3
Fractional Exponents
Example E.a. 16½ = 16 = 4b. 9½ 25½ =925 = 3*5 = 15c. 43/2 = (4½ * 3) = (4½)3 = (4)3 = 23 = 8
d. 16 – ½ = (16½) –1 = (16) –1 = 4 –1 = 1/4e. 9 –3/2 = (9 –½ * 3) = (9 ½) –3 = (9)–3 = 3–3 = 1/27
A–1/2 = andA1
1/k-Power Rule A1/k = Ak
A –k/2 = ( )k
A1
Using an argument similar to the argument for ½-power rule, we have the following rule for 1/k power.
In general,
Example F. Simplify.a. 81/3 = 8 = 2b. 27 –2/3 = (271/3)–2
3
Fractional Exponents
Example E.a. 16½ = 16 = 4b. 9½ 25½ =925 = 3*5 = 15c. 43/2 = (4½ * 3) = (4½)3 = (4)3 = 23 = 8
d. 16 – ½ = (16½) –1 = (16) –1 = 4 –1 = 1/4e. 9 –3/2 = (9 –½ * 3) = (9 ½) –3 = (9)–3 = 3–3 = 1/27
A–1/2 = andA1
1/k-Power Rule A1/k = Ak
A –k/2 = ( )k
A1
Using an argument similar to the argument for ½-power rule, we have the following rule for 1/k power.
In general,
Example F. Simplify.a. 81/3 = 8 = 2b. 27 –2/3 = (271/3)–2 = (27)–2
3
3
Fractional Exponents
Example E.a. 16½ = 16 = 4b. 9½ 25½ =925 = 3*5 = 15c. 43/2 = (4½ * 3) = (4½)3 = (4)3 = 23 = 8
d. 16 – ½ = (16½) –1 = (16) –1 = 4 –1 = 1/4e. 9 –3/2 = (9 –½ * 3) = (9 ½) –3 = (9)–3 = 3–3 = 1/27
A–1/2 = andA1
1/k-Power Rule A1/k = Ak
A –k/2 = ( )k
A1
Using an argument similar to the argument for ½-power rule, we have the following rule for 1/k power.
In general,
Example F. Simplify.a. 81/3 = 8 = 2b. 27 –2/3 = (271/3)–2 = (27)–2
3
3
Fractional Exponents
Example E.a. 16½ = 16 = 4b. 9½ 25½ =925 = 3*5 = 15c. 43/2 = (4½ * 3) = (4½)3 = (4)3 = 23 = 8
d. 16 – ½ = (16½) –1 = (16) –1 = 4 –1 = 1/4e. 9 –3/2 = (9 –½ * 3) = (9 ½) –3 = (9)–3 = 3–3 = 1/27
A–1/2 = andA1
1/k-Power Rule A1/k = Ak
A –k/2 = ( )k
A1
Using an argument similar to the argument for ½-power rule, we have the following rule for 1/k power.
In general,
Example F. Simplify.a. 81/3 = 8 = 2b. 27 –2/3 = (271/3)–2 = (27)–2 = (3)–2
3
3
Fractional Exponents
Example E.a. 16½ = 16 = 4b. 9½ 25½ =925 = 3*5 = 15c. 43/2 = (4½ * 3) = (4½)3 = (4)3 = 23 = 8
d. 16 – ½ = (16½) –1 = (16) –1 = 4 –1 = 1/4e. 9 –3/2 = (9 –½ * 3) = (9 ½) –3 = (9)–3 = 3–3 = 1/27
A–1/2 = andA1
1/k-Power Rule A1/k = Ak
A –k/2 = ( )k
A1
Using an argument similar to the argument for ½-power rule, we have the following rule for 1/k power.
In general,
Example F. Simplify.a. 81/3 = 8 = 2b. 27 –2/3 = (271/3)–2 = (27)–2 = (3)–2 = 1/32 = 1/9
3
3
Fractional Exponents
All the following rules for operations of exponents hold for fractional exponents N and K.
Fractional Exponents
All the following rules for operations of exponents hold for fractional exponents N and K.
Multiplication Rule ANAK = AN + K
Fractional Exponents
Division Rule = AN – K
Power Rule (AN)K = ANK
All the following rules for operations of exponents hold for fractional exponents N and K.
AN
AK
Multiplication Rule ANAK = AN + K
Fractional Exponents
Division Rule = AN – K
Power Rule (AN)K = ANK
All the following rules for operations of exponents hold for fractional exponents N and K.
x(x1/3y3/2)2
x–1/2y2/3
AN
AK
Example G. Simplify the exponents.
Multiplication Rule ANAK = AN + K
Fractional Exponents
Division Rule = AN – K
Power Rule (AN)K = ANK
All the following rules for operations of exponents hold for fractional exponents N and K.
x(x1/3y3/2)2
=x*x2/3y3
x–1/2y2/3
2*1/3 2*3/2
AN
AK
Example G. Simplify the exponents.
x–1/2y2/3
Multiplication Rule ANAK = AN + K
Fractional Exponents
Division Rule = AN – K
Power Rule (AN)K = ANK
All the following rules for operations of exponents hold for fractional exponents N and K.
x(x1/3y3/2)2
=x*x2/3y3
x–1/2y2/3=
2*1/3 2*3/2
x5/3y3
1+2/3
AN
AK
Example G. Simplify the exponents.
x–1/2y2/3 x–1/2y2/3
Multiplication Rule ANAK = AN + K
Fractional Exponents
Division Rule = AN – K
Power Rule (AN)K = ANK
All the following rules for operations of exponents hold for fractional exponents N and K.
x(x1/3y3/2)2
=x*x2/3y3
x–1/2y2/3=
2*1/3 2*3/2
=
x5/3y3
1+2/3
x5/3 – (–1/2) y3 – 2/3
AN
AK
Example G. Simplify the exponents.
x–1/2y2/3 x–1/2y2/3
Multiplication Rule ANAK = AN + K
Fractional Exponents
Division Rule = AN – K
Power Rule (AN)K = ANK
All the following rules for operations of exponents hold for fractional exponents N and K.
x(x1/3y3/2)2
=x*x2/3y3
x–1/2y2/3=
2*1/3 2*3/2
=
x5/3y3
1+2/3
x5/3 – (–1/2) y3 – 2/3
AN
AK
Example G. Simplify the exponents.
x–1/2y2/3 x–1/2y2/3
= x5/3 +1/2 y7/3
Multiplication Rule ANAK = AN + K
Fractional Exponents
Division Rule = AN – K
Power Rule (AN)K = ANK
All the following rules for operations of exponents hold for fractional exponents N and K.
x(x1/3y3/2)2
=x*x2/3y3
x–1/2y2/3=
2*1/3 2*3/2
=
x5/3y3
1+2/3
x5/3 – (–1/2) y3 – 2/3
= x13/6 y7/3
AN
AK
Example G. Simplify the exponents.
x–1/2y2/3 x–1/2y2/3
= x5/3 +1/2 y7/3
Multiplication Rule ANAK = AN + K
Fractional Exponents
Exercise A. Write the expressions in radicals and simplify.
1. 41/2 2. 91/2 3. 161/2 4. 251/2
6. 361/2 7. 491/2 8. 641/2 9. 811/2
10. (100x)1/2x4( )1/211.
925( )1/212.
13. (–8)1/3 14. –164( )1/3
16. 4–1/2 17. 9–1/2 18. 16–1/2 19. 25–1/2
20. 36–1/2 21. 49–1/2 22. 64–1/2 23. 81–1/2
15. 1251/3
x4( ) –1/225.
925( ) –1/226.
27. –164( ) –1/3 28. 125–1/3
x(24. )–1/2100
29. 163/2 30. 253/2
31. –164( ) –2/3 32. 125–2/3
Fractional Exponents
Exercise B. Simplify the expressions by combining the exponents. Interpret the problems and the results in radicals.
34. x1/2x1/4 33. x1/3x1/3 35. x1/2x1/3
37. 36. 38. x1/3
x1/4 x1/3 x1/3x1/2
Fractional Exponents
x1/2
Exercise C. Simplify the expressions by combining the exponents. Leave the answers in simplified fractional exponents.
40. x1/4x3/2x2/3 39. x1/3x4/3x1/2
45.
48. y1/2x1/3x–1/4
x4/3y–3/2
x1/2
41. x3/4y1/2x3/2y4/3
43. y–2/3x–1/4y–3/2x–2/3 42. y–1/4x5/6x4/3y1/2 44. x3/4y1/2x3/2y4/3
46. x1/4 x–1/2
47. x–1/4 x–1/2
49. y–1/2x1/3
y–4/3x–3/250. (8x2)1/3
(4x)3/2
51. (27x2)–1/3
(9x)1/252. (100x2)–1/2
(64x)1/353. (36x3)–1/2
(64x1/2)–1/3
Related Documents
