¸¡5.pdf · n ของจํานวนจริง. บทนิยาม ให . x . และ . y . เป นจํานวนจริง และ . n . เป นจํา

เอกสารประกอบกิจกรรมการเรียนนวัตกรรมชั้นเคล่ือน ม.5 สัปดาหท่ี


สัญลักษณของรากที่ n ของจํานวนจริง

บทนิยาม ให x และ y เปนจํานวนจริง และ n เปนจํานวนเต็มท่ีมากกวา 1 y เปนคาหลักของรากท่ี x

เขียนแทนดวย n x ก็ตอเมื่อ

1) y เปนรากท่ี n ของ x

2) xy≥ 0

สําหรับ n x อานวา กรณฑท่ี n ของ x หรือคาหลักของรากท่ี n ของ x

นารู!!!

1.สัญลักษณ เรียกวา เคร่ืองหมายกรณฑ (radical sign)

2. รากที่ n เมื่อ n เปนจํานวนเต็มที่มากกวา 1 เขียนแทนดวย n และเรียก n วา

อันดับที่หรือดัชนี (index) ของกรณฑ

3. กรณี กรณฑที่ 2 ( n = 2) เขียนแทนดวย

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ตรวจสอบความรู

ใหนักเรียนเขียนสรุปความรูที่ไดจากใบความรู (นักเรียนสามารถคนหาเพ่ิมเติมได)

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2


ใบงานท่ี 1

ใหนักเรียนสรุปความรูที่ไดจากใบความรูเร่ืองสัญลักษณของรากที่ n ของจํานวนจริง

เปนผังความคดิพรอมทั้งตกแตงใหสวยงาม

2


การเปลี่ยนประโยคภาษาเปนประโยคสัญลักษณ

การเปลี่ยนประโยคภาษาเปนประโยคสัญลักษณของรากที่ n ของจํานวนจริง น้ันทําได ดังตัวอยาง

ตัวอยางจงเขียนคําอานของกรณฑตอไปน้ี

1. 3 2 อานวา รากที่สามของสอง

2. 5 15- อานวา รากที่หาของลบสิบหา

3. n y อานวา รากที่ n ของ y

4. 8 อานวา กรณฑที่สองของแปดที่เปนบวก

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① จงเขียนคําอานของกรณฑตอไปน้ี

1.1 6 258 อานวา ...............................................................................................

1.2 89 อานวา ...............................................................................................

1.3 4 16- อานวา ...............................................................................................

② จงเขียนประโยคสัญลักษณจากประโยคตอไปน้ี

2.1 กรณฑสามของหารอยยี่สิบสี่ ประโยคสัญลักษณคือ .....................................................

2.2 กรณฑที่สองของยี่สิบหก ประโยคสัญลักษณคือ .....................................................

2.3 กรณฑที่เจ็ดของลบสามรอยหกสิบ ประโยคสัญลักษณคือ .......................................

3



ใหนักเรียนเติมคําตอบใหถูกตอง

❶จงเขียนคําอานของกรณฑตอไปน้ี

1.1 3 25 อานวา ....................................................................................................

1.2 5 6 อานวา ....................................................................................................

1.3 8- อานวา ....................................................................................................

1.4 2

)8(- อานวา ....................................................................................................

1.5 4 125 อานวา ....................................................................................................

❷จงเขียนประโยคสัญลักษณจากประโยคตอไปน้ี

2.1 รากที่สองของสามรอย ประโยคสัญลักษณคือ .............................................

2.2 รากที่สี่ของลบยี่สิบแปด ประโยคสัญลักษณคือ .............................................

2.3 กรณฑที่สามของสิบหา ประโยคสัญลักษณคือ .............................................

2.4 กรณฑที่สองของลบสิบแปด ประโยคสัญลักษณคือ .............................................

2.5 ลบกรณฑที่สองของลบหา ประโยคสัญลักษณคือ .............................................

3

4


สมบัติพ้ืนฐานของรากที่ n ของจํานวนจริง

ในกรณีทั่วไปมีขอสรุปเกี่ยวกับคาหลักของรากที่ n ของจํานวนจริง x หรือ n x ดังน้ี

1. ถา x = 0 แลว n x = 0

2. ถา x > 0 แลว n x เปนจํานวนจริงบวก

3. ถา x < 0 แลว n x เปนจํานวนจริงลบ

ในคร้ังน้ีจะยกตัวอยางของสมบัติพื้นฐานของรากที่ n ในขอ 1. และ 2. สําหรับในขอที่ 3. จะยกตัวอยาง

ในคร้ังถัดไป

ตัวอยางจงหาคาของ

1. 1) 3 0 2) 5 32 3) 4 81

วิธีทํา 1) เน่ืองจาก 03= 0 ดังน้ัน 3 0 = 0

2) เน่ืองจาก 25= 32 ดังน้ัน 5 32 = 2

3) เน่ืองจาก 34= 81 ดังน้ัน 4 81 = 3

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จงหาคาของ

1) 4 243

...................................................................

...................................................................

2) 3 27

...................................................................

...................................................................

3) 16

...................................................................

...................................................................

4) 5 0

...................................................................

...................................................................





1. 3 216

...........................................................................

...........................................................................

2. 521

...........................................................................

...........................................................................

3. 5 32

...........................................................................

...........................................................................

4. 1600

...........................................................................

...........................................................................

5. 3 125

...........................................................................

...........................................................................

6. 4 16

...........................................................................

...........................................................................

7. 3 729

...........................................................................

...........................................................................

8. 4 625

...........................................................................

...........................................................................

9. 000,10

...........................................................................

...........................................................................

10. 3 343

...........................................................................

...........................................................................

4


สมบัติพ้ืนฐานของรากที่ n ของจํานวนจริง

ในกรณีทั่วไปมีขอสรุปเกี่ยวกับคาหลักของรากที่ n ของจํานวนจริง x หรือ n x ดังน้ี

1. ถา x = 0 แลว n x = 0

2. ถา x > 0 แลว n x เปนจํานวนจริงบวก

3. ถา x < 0 และ n เปนจํานวนคี่ แลว n x เปนจํานวนจริงลบ

ในคร้ังน้ีจะยกตัวอยางของสมบัติพื้นฐานของรากที่ n ในขอที่ 3. (สําหรับ 1. และ 2. กลาวถึงในคร้ังกอนหนา )


1) 5 32- 2) 3 27- 3) 3 8-

วิธีทํา 1) เน่ืองจาก (-2)5= -32 ดังน้ัน 5 32- = -2

2) เน่ืองจาก (-3)3 = -27 ดังน้ัน 3 27- = -3

3) เน่ืองจาก (-2)3= -8 ดังน้ัน 3 8- = -2

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1) 3 216-

...................................................................

...................................................................

2) 5 32-

...................................................................

...................................................................

3) 3 125-

...................................................................

...................................................................

4) 5 128-

...................................................................

...................................................................

5





1. 3 216-

...........................................................................

...........................................................................

2. 5 243-

...........................................................................

...........................................................................

3. 5 32-

...........................................................................

...........................................................................

4. 3 64-

...........................................................................

...........................................................................

5. 3 125-

...........................................................................

...........................................................................

6. 7 128-

...........................................................................

...........................................................................

7. 3 729-

...........................................................................

...........................................................................

8. 3 125-

...........................................................................

...........................................................................

9. 5 100,000-

...........................................................................

...........................................................................

10. 3 343-

...........................................................................

...........................................................................

5


สมบัตทิี่สําคัญของรากที่ n (1)

สมบัติของรากที่ n

ให a และ b เปนจํานวนจริงที่มีรากที่ n และ n เปนจํานวนเต็มที่มากกวา 1

1. nn )a( = a เมื่อ n a เปนจํานวนจริง

a เมื่อ a ≥ 0

2. n na = a เมื่อ a < 0 และ n เปนจํานวนคี่บวก

a เมื่อ a < 0 และ n เปนจํานวนคูบวก

3. n ab = nn b•a

4. n

b

a =

n

n

b

a เมื่อ b≠ 0

ในคร้ังน้ีจะยกตัวอยางของสมบติัสําคัญของรากที่ n ในขอ 1. และ 2. สําหรับในขอที่ 3. จะยกตัวอยาง

ในคร้ังถัดไป

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จงยกตัวอยางรากที่ n โดยใชสมบัติขอ 1. และ ขอ 2 มา 2 ขอ

1)

...................................................................

...................................................................

...................................................................

...................................................................

2)

...................................................................

...................................................................

...................................................................

...................................................................

6





1. 33 )216(

...........................................................................

...........................................................................

2. 55 )243-(

...........................................................................

...........................................................................

3. 55 )32-(

...........................................................................

...........................................................................

4. 2)64-(

...........................................................................

...........................................................................

5. 3 3)5(-

...........................................................................

...........................................................................

6. 4 42

...........................................................................

...........................................................................

7. 3 3)729(-

...........................................................................

...........................................................................

8. 33 )125-(

...........................................................................

...........................................................................

9. 2)100-(

...........................................................................

...........................................................................

10. 3 3)343(-

...........................................................................

...........................................................................

6









3. n ab = nn b•a

4. n

b

a =

n

n

b


ในคร้ังน้ีจะยกตัวอยางของสมบัติสําคัญของรากที่ n ในขอ 3. (สําหรับในขอที่ 1. และ 2.ไดยกตัวอยางใน

คร้ังกอนหนา)

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จงยกตัวอยางรากที่ n โดยใชสมบัติขอ 3. มา 2 ขอ

1)

...................................................................

...................................................................

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2)

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7





1. 3 23 x•x

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2. 4 24 22•2

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3. )4)(8-( 55

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4. )4)(64(

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...........................................................................

5. 3 3)5(-

...........................................................................

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6. 4 42

...........................................................................

...........................................................................

7. 3 3)729(-

...........................................................................

...........................................................................

8. 33 )125-(

...........................................................................

...........................................................................

9. 2)100-(

...........................................................................

...........................................................................

10. 3 3)343(-

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...........................................................................

7


การหาคาของ n na

เมื่อ a ≥ 0

จากสมบติัของรากที่ n


a เมื่อ a ≥ 0 n n

a = a เมื่อ a < 0 และ n เปนจํานวนคี่บวก


ตัวอยาง จงหาคาของ

1) 5 52 2) 3 3

4 3) 4 256 4) 36

วิธีทํา 1) 5 52 = 2 2) 3 3

4 = 4

3) 4 256 = 4 44 = 4 4) 36 = 2

6 = 6

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จงหาคาของรากที่ n ตอไปน้ี

1) 3 33

...................................................................

...................................................................

2) 4 625

...................................................................

...................................................................

3) 5 50

...................................................................

...................................................................

4) 5 216

...................................................................

...................................................................

8





1. 3 001.0

...........................................................................

...........................................................................

2. 3 3375

...........................................................................

...........................................................................

3. )32(5

...........................................................................

...........................................................................

4. 64

...........................................................................

...........................................................................

5. 3 35

...........................................................................

...........................................................................

6. 4 42

...........................................................................

...........................................................................

7. 3 729

...........................................................................

...........................................................................

8. )125(3

...........................................................................

...........................................................................

9. )100(

...........................................................................

...........................................................................

10. 3 343

...........................................................................

...........................................................................

8


การหาคาของ n na

เมื่อ a < 0

จากสมบัติของรากที่ n

ให a เปนจํานวนจริงที่มีรากที่ n และ n เปนจํานวนเต็มที่มากกวา 1





1) 5 5)2(- 2) 3 3

)4(- 3) 3 8- 4) 5 32 -

วิธีทํา 1) 5 5)2(- = -2 2) 3 3

)4(- = -4

3) 3 8- = 3 3)2(- = -2 4) 5 32 - = 5 5

2) (- = -2

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จงหาคาของรากที่ n ตอไปน้ี

1) 3 3)3(-

...................................................................

...................................................................

2) 7 7)6(-

...................................................................

...................................................................

3) 5 243-

...................................................................

...................................................................

4) 5 216-

...................................................................

...................................................................

9





1. 3 001.0-

...........................................................................

...........................................................................

2. 3 3375-

...........................................................................

...........................................................................

3. )32-(5

...........................................................................

...........................................................................

4. 3 512-

...........................................................................

...........................................................................

5. 3 3)5(-

...........................................................................

...........................................................................

6. 3 3)8(-

...........................................................................

...........................................................................

7. 3 729-

...........................................................................

...........................................................................

8. 3 125-

...........................................................................

...........................................................................

9. )1728-(3

...........................................................................

...........................................................................

10. 3 343-

...........................................................................

...........................................................................

9


การหาคาของรากที่สอง

จากสมบัติของรากที่ 2

กําหนดจํานวนจริงที่มีรากที่ 2 และ a เปนจํานวนเต็มที่มากกวาหรือเทากับ 0

n

a = a


1) 42 2) 2

36 3) 25

วิธีทํา 1) 42 = 22 = 4

2) 236 = 36

3) 25 = 5

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จงหาคาของรากที่ 2 ตอไปน้ี

1) 0

...................................................................

...................................................................

2) 66

...................................................................

...................................................................

3) 16

...................................................................

...................................................................

4) 625

...................................................................

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10





1. 01.0

...........................................................................

...........................................................................

2. 0

...........................................................................

...........................................................................

3. )64(

...........................................................................

...........................................................................

4. 625

...........................................................................

...........................................................................

5. 100

...........................................................................

...........................................................................

6. )16(4

...........................................................................

...........................................................................

7. 096,4

...........................................................................

...........................................................................

8. 401,2

...........................................................................

...........................................................................

9. 21,728

...........................................................................

...........................................................................

10. 81

...........................................................................

...........................................................................

10


การหาคารากที่ n

ถา 𝐚 ≥ 𝟎 และ 𝐧 ≥ 𝟎

จากสมบติัของรากที่ n

กําหนดจํานวนจริงที่มีรากที่ n เมื่อ a และ n เปนจํานวนเต็มที่มากกวาหรือเทากับ 0




ตัวอยาง


1) 40 2) 0

36 3) 25

วิธีทํา 1) 40 = 0

2) 036 = 1

3) 81 = 9

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1) 42

...................................................................

...................................................................

2) 60

...................................................................

...................................................................

3) 05

...................................................................

...................................................................

4) 36

...................................................................

...................................................................

11




จงทําจํานวนตอไปน้ีใชอยูในรูปอยางงาย

1. 500

...........................................................................

...........................................................................

2. 16

...........................................................................

...........................................................................

3. 500,2

...........................................................................

...........................................................................

4. 648

...........................................................................

...........................................................................

5. 64

...........................................................................

...........................................................................

6. 18

...........................................................................

...........................................................................

7. 096,4

...........................................................................

...........................................................................

8. 401,2

...........................................................................

...........................................................................

9. 1,728(

...........................................................................

...........................................................................

10. 81

...........................................................................

...........................................................................

11


การหาคารากที่ n

ถา 𝐚 < 𝟎 เมื่อ 𝐚 𝝐 𝑹

จากสมบัติของรากที่ n

กําหนดจํานวนจริงที่มีรากที่ n เมื่อ a และ n เปนจํานวนเต็มทีม่ากกวาหรือเทากบั 0




ตัวอยาง


1) 4(-3) 2) 2

(-36) 3) 2

)25(-

วิธีทํา 1) 4(-81) = 9

2) 2(-36) = 36

3) 2

)25(- = 25

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1) 4)2(-

...................................................................

...................................................................

2) 2(-8)

...................................................................

...................................................................

3) 2(-7)

...................................................................

...................................................................

4) 4)5(-

...................................................................

...................................................................

12





1. 2)216(

...........................................................................

...........................................................................

2. 2)243-(

...........................................................................

...........................................................................

3. 4)32(-

...........................................................................

...........................................................................

4. 2)(-64

...........................................................................

...........................................................................

5. 2)5(-

...........................................................................

...........................................................................

6. 4)2(-

...........................................................................

...........................................................................

7. 2)729(-

...........................................................................

...........................................................................

8. 2)125(-

...........................................................................

...........................................................................

9. 4(-10)

...........................................................................

...........................................................................

10. 2)343(-

...........................................................................

...........................................................................

12









7. n ab = nn b•a

8. n

b

a =

n

n

b


ในคร้ังน้ีจะยกตัวอยางของสมบัติสําคัญของรากที่ n ในขอ 3. (สําหรับในขอที่ 1. และ 2.ไดยกตัวอยางใน

คร้ังกอนหนา)

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จงยกตัวอยางรากที่ n โดยใชสมบัติขอ 3. มา 2 ขอ

2)

...................................................................

...................................................................

...................................................................

...................................................................

2)

...................................................................

...................................................................

...................................................................

...................................................................

13




จงทําใหอยูในรูปอยางงายโดยท่ีตัวสวนไมติดกรณฑ

1. 2+3

2

...........................................................................

...........................................................................

...........................................................................

...........................................................................

2. 5

5

...........................................................................

...........................................................................

...........................................................................

...........................................................................

3. 8

62

...........................................................................

...........................................................................

...........................................................................

...........................................................................

4. 22-3

4

...........................................................................

...........................................................................

...........................................................................

...........................................................................

5. 2+75

8-3

...........................................................................

...........................................................................

...........................................................................

...........................................................................

6. 2-75

8+3

...........................................................................

...........................................................................

...........................................................................

...........................................................................

13


กรณฑที่เหมือนกนั

กรณฑที่เหมือนกัน หมายถึง ..................................

- มีอันดับของกรณฑเปนอันดับเดียวกัน

- จํานวนที่อยูภายใตกรณฑ เปนจํานวนเดียวกัน

ตัวอยางจงจับคูกรณฑที่เหมือนกันใหถูกตอง

1) 52 2) 3 6256 3) 4 32

ก) 3 5 ข) 625 ค) 4 2 -----------------------------------------------------------------------------------------------


จงเขียน หนาขอที่ถูก และเขียน หนาขอที่ผิด

............... 1) 5 2 เปนกรณฑที่เหมือนกันกับ 5 64

............... 2) 8 เปนกรณฑที่เหมือนกันกับ 5 8




14




จงเขียนกรณฑท่ีเหมือนกันกับกรณฑตอไปน้ีมา 3 ตัวอยาง

1. 3 8 2. 365 3. 4 000,1

.................................................... .................................................... ....................................................

.................................................... .................................................... ....................................................

.................................................... .................................................... ....................................................

4. 3 128 5. 200 6. 4 200,1

.................................................... .................................................... ....................................................

.................................................... .................................................... ....................................................

.................................................... .................................................... ....................................................

7. 3 135 8. 108 9. 3 54

.................................................... .................................................... ....................................................

.................................................... .................................................... ....................................................

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14


การบวกจํานวนเต็มทีอ่ยูในรูปกรณฑ

การหาผลบวกและผลตางของจํานวนจริงที่มีเคร่ืองหมายกรณฑอันดับเดียวกนั

และมีจํานวนภายในกรณฑเปนจํานวนเดียวกัน

ตัวอยางจงเขียนจํานวนตอไปน้ีในรูปอยางงาย

1) 50+32 = 2×25+2×16

= 2•25+2•16

= 25+24

= 29

2) 4 54 16x+x81 = ( )4 44 4x2x+x3

= 44 x2x+x3

= 4 x)x2+3(

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จงเขียนจํานวนตอไปน้ีในรูปอยางงาย

1) 108+75 2)5 65 243x+x32

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15




จงทําใหอยูในรูปอยางงาย

1. 3 43 2xy27+xy3 2. 80+45+20

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3. 3 43 2xy27+y 4. 80+45+20

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5. x4+x49 6. 33 2+250

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15


การบวกจํานวนเต็มและเศษสวนที่อยูในรูปกรณฑ

การหาผลบวกและผลตางของจํานวนจริงที่มีเคร่ืองหมายกรณฑอันดับเดียวกัน



3) 5

50+32 =

5

50+2×16

= 10+2•16

= 10+24

4) 4 5

3

4

16x+81

x81 = ( )4 4

3

4 4

x2x+81

x3

= 3

x6x+

3

x3 44

= 3

x6x+x3 44

= 44 x2x+x

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ตรวจสอบความรู จงเขียนจํานวนตอไปน้ีในรูปอยางงาย

2) 2

108+75 2)

5

55 5

x

x32+243x

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16



ใหนักเรียนเติมคําตอบใหถูกตอง จงทําใหอยูในรูปอยางงาย

1. 20+245 2. 316+27+34

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3. 33 5+125 4. 4

4

4

1024+2

32

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5. 6

x36+x25

42

6. 3 3

3

3 6

y27+7

y2401

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16


การบวกเศษสวนและทศนิยมที่อยูในรูปกรณฑ




1)5

50+

5

32 =

5

2×25+2×16

= 5

2•25+2•16

= 5

25+24

= 5

29

3) 50.16x+x81.0 = xx0.4+x9.0

422

= x0.4x+x9.02

= x)0.4x+9.0(2

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1)5

108+75 2) 3

0.64y+0.01y

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17



ใหนักเรียนเติมคําตอบใหถูกตอง จงทําใหอยูในรูปอยางงาย

1. 42x36.0+x25.0 2. 316+27+

5

34

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3. 33 5.0+0625.0 4. 4

4

4

1024+2

32

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5. 6.0

x216.0+x25.0

42

6. 3 3

3

3 6

y27+7

y2401

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17


การลบจํานวนเต็มทีอ่ยูในรูปกรณฑ




1) 50-32 = 2×25-2×16

= 2•25-2•16

= 25-24

= 2-

2) 4 54 16x-x81 = ( )4 44 4x2x-x3

= 44 x2x-x3

= 4 x)x2-3(

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1) 108-75 2) 5 65 243x-x32

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1. 20-245 2. 273-34

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3. 2427a-12a 4. 278-37

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5. 42x36-x25 6. 3 33 6

y27-y343

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18


การลบเศษสวนและทศนิยมที่อยูในรูปกรณฑ




1) 5

50-

5

32 =

5

2×25-2×16

= 5

2•25-2•16

= 5

25-24

= 5

2-

2) 50.16x-x81.0 = xx0.4-x9.0

422

= x0.4x-x9.02

= x)0.4x-9.0(2

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1) 5

108-75 2) 3

0.64y-0.01y

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1. 500

20-245 2.

3

2-34

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3. 3

a4-a12-

3

a 4.

3

8-37

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5. 3

x36.0-

3

x25.042

6. 3 33 6

y27-5

y343

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การบวกและลบจํานวนเต็มที่อยูในรูปกรณฑ




1) 48-50+32 = -2×25+2×16 3×42

= 34-2•25+2•16

= 34-25+24

= 34-29

2) 44 54 x+16x-x81 = ( ) 44 44 4x+x2x-x3

= 444 x+x2x-x3

= 4 x)1+x2-3(

= 4 x)x2-4(

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1) 125+108-75 2) 55 65 x+243x-x32

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20





1. 20-245+5 2. 485-34+753

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3. 3

a4-a12+a 4. 243+275-37

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5. 802-459+207 6. 33 33 66561+y27-y343

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20


การบวกและลบเศษสวนและทศนิยมทีอ่ยูในรูปกรณฑ




1) 2

3+22 = )

2

2×

2

3(+22

= 2

23+22

= 2

23+

2

24

= 2

27

2) 10.24x-x84.7 = x3.2-x8.2

= x3.2)-8.2(

= x0.4-

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1) 3

6-

3

10+312 2) y+0.64y-0.01y

3

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1. 5+500

20-245 2.

3

2-34+53

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3. 3

a4-a12+

3

a 4.

1+2

8+

3

8-37

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5. 2

3+

3

x36.0-

3

x25.042

6. 33 33 6

6561+y27-5

y343

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21


การบวกและการลบจํานวนเต็ม

เศษสวนและทศนยิมที่อยูในรูปกรณฑ

ใหนักเรียนตั้งโจทยเกีย่วกับการบวกและการลบจํานวนเต็ม เศษสวนและทศนิยมที่

อยูในรูปกรณฑ จํานวน 5 ขอ

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22



ใหนักเรียนจับคูกับเพ่ือนแลวสลับกันทําโจทยที่เพ่ือนตั้ง จํานวน 5 ขอ

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22


การเปลี่ยนเลขยกกําลังใหอยูในรูปของราก

บทนิยาม ถา a เปนจํานวนจริง m, n เปนจํานวนเต็มท่ีn > 1

และ n

m เปนเศษสวนอยางต่ํา และ n

1

a เปนจํานวนจริงจะไดวา

mnmn

1

n

m

)a(=)a(=a

n mn

1mn

m

a=)a(=a

ตัวอยางจงเขียนใหอยูในรูปกรณฑ

1) 3

2

2 = 3 22

2) 4

5

)16( = 4 516

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จงเขียนใหอยูในรูปกรณฑ

1) 3

1

54 2) 5

3

)75(-

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3) 2

3

93 4) 2

3

15

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23




จงหาเขียนใหอยูในรูปกรณฑหรือทําใหอยูในรูปอยางงาย

1. 3

2

8 2. 5

2-

)-32(

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3. 2

3

)-2( 4. ))125(( 2

5

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5. 2

6

)-28( 6. 5

8

64

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23


การคูณกรณฑดวยกรณฑ

สมบัติ ให a และ b เปนจํานวนจริงท่ีมีรากท่ี n และ n เปนจํานวนเต็มท่ีมากกวา 1

nnn b•a=ab


1) 6•3•2

= 6×3×2

= 6×6

= 6

2) 333 9•25•15

= 333 3×3•5×5•5×3

= 33 5×5×5•3×3×3

= 3×5

= 15

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จงหาคาของเลขยกกําลังตอไปน้ี

1) 3•5•15 2) 333 9•4•6 3) )2-4)(2+3(

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จงหาคาของจํานวนตอไปน้ี

1. 50•15•3 2. 333 16•88•26

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3. 66•18•56 4. 444 54•69•5

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5. 33 49•7- 6. 3•108

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24


การเปลี่ยนขนาดของอนัดับกรณฑ

ให a และ b เปนจํานวนจริงท่ีมีรากท่ี n และ n แล k เปนจํานวนเต็มท่ีมากกวา 1

nk kn a=a


1) 3•23

= 3•2 32•3 23•2

= 6 36 23•2

= 6 108

2) 43 2•2

= 3•4 34•3 4

2•2

= 6 34

2×2

= 6 6

2×2

= 6 22

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จงหาคาของเลขยกกําลังตอไปน้ี

1) 3 42•2 2) 43 4•6 3) 35 625•25

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1. 8•263 2. 3 26•12

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3. 6•44 4. 34 625•5

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25


การสังยุค


1) 1+2

1

= 1-2

1-2×

1+2

1

= 22

1-)2(

1-2

= 1-2

1-2

= 1-2

2) 2-5

2+5

= 2+5

2+5×

2-5

2+5

= 22

22

)2(-)5(

)2(+252+)5(

= 2-5

2+102+5

= 3

102+7

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1) 2-3

2+3 2)

1+2

1-2

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1. 3+2

8 2.

3+5

3-5

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3. 3-7

6+6 4.

5-22

22+5

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26


การหารกรณฑดวยกรณฑ

ให a และ b เปนจํานวนจริงท่ีมีรากท่ี n และ n เปนจํานวนเต็มท่ีมากกวา 1

n

n

n

b

a=

b

a


1) 3

3

4

160-

= 3

4

160-

= 3 8-

= -2

2) 2-5

2+5

= 2+5

2+5×

2-5

2+5

= 22

22

)2(-)5(

)2(+252+)5(

= 2-5

2+102+5

= 3

102+7

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1) 5

5

2

2048 2)

1+23

3-2

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27





1. 5

3

3

729 2.

3

3

6

1296

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3. 33-72

5+2 4.

5+22

5-22

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Documents

¸¡5.pdf · n ของจํานวนจริง. บทนิยาม ให . x . และ . y . เป นจํานวนจริง และ . n . เป นจํา