ГДЗ ответы к учебнику Алгебра 8 класс, Мерзляк, Полонский, Якир - Ответы к странице 44 ГДЗ к учебнику алгебры Мерзляк, 8 класс

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Ответы к странице 44

177. Упростите выражение:
1) (x+xy):(x−xy)(x+xy):(x−xy);
2) (ab+a+ba−b)∗ab2a2+b2(ab+a+ba−b)∗ab2a2+b2;
3) (mm−1−1):mmn−n(mm−1−1):mmn−n;
4) (ab−ba)∗4aba−b(ab−ba)∗4aba−b;
5) ab−a2−b2b2:a+bbab−a2−b2b2:a+bb;
6) 7xx+2−x−83x+6∗84x2−8x7xx+2−x−83x+6∗84x2−8x;
7) (a−9a−9a+3):a2−3aa+3(a−9a−9a+3):a2−3aa+3;
8) (aa+2−8a+8)∗a2+8aa−4(aa+2−8a+8)∗a2+8aa−4.

Решение:

1) (x+xy):(x−xy)=xy+xy:xy−xy=x(y+1)y:x(y−1)y=x(y+1)y∗yx(y−1)=y+11∗1y−1=y+1y−1(x+xy):(x−xy)=xy+xy:xy−xy=x(y+1)y:x(y−1)y=x(y+1)y∗yx(y−1)=y+11∗1y−1=y+1y−1

2) (ab+a+ba−b)∗ab2a2+b2=a(a−b)+b(a+b)b(a−b)∗ab2a2+b2=a2−ab+ab+b2b(a−b)∗ab2a2+b2=a2+b2b(a−b)∗ab2a2+b2=1a−b∗ab1=aba−b(ab+a+ba−b)∗ab2a2+b2=a(a−b)+b(a+b)b(a−b)∗ab2a2+b2=a2−ab+ab+b2b(a−b)∗ab2a2+b2=a2+b2b(a−b)∗ab2a2+b2=1a−b∗ab1=aba−b

3) (mm−1−1):mmn−n=m−(m−1)m−1:mn(m−1)=m−m+1m−1:mn(m−1)=1m−1∗n(m−1)m=11∗nm=nm(mm−1−1):mmn−n=m−(m−1)m−1:mn(m−1)=m−m+1m−1:mn(m−1)=1m−1∗n(m−1)m=11∗nm=nm

4) (ab−ba)∗4aba−b=a∗a−b∗bab∗4aba−b=a2−b21∗4a−b=(a−b)(a+b)1∗4a−b=a+b1∗41=4(a+b)

5) ab−a2−b2b2:a+bb=ab−(a−b)(a+b)b2∗ba+b=ab−a−bb∗11=a−(a−b)b=a−a+bb=bb=1

6) 7xx+2−x−83x+6∗84x2−8x=7xx+2−x−83(x+2)∗84x(x−8)=7xx+2−1x+2∗28x=7xx+2−28x(x+2)=7x∗x−28x(x+2)=7x2−28x(x+2)=7(x2−4)x(x+2)=7(x−2)(x+2)x(x+2)=7(x−2)x

7) (a−9a−9a+3):a2−3aa+3=a(a+3)−(9a−9)a+3:a(a−3)a+3=a2+3a−9a+9a+3∗a+3a(a−3)=a2−6a+91∗1a(a−3)=(a−3)2a(a−3)=a−3a

8) (aa+2−8a+8)∗a2+8aa−4=a(a+8)−8(a+2)(a+2)(a+8)∗a(a+8)a−4=a2+8a−8a−16a+2∗aa−4=a2−16a+2∗aa−4=(a−4)(a+4)a+2∗aa−4=a+4a+2∗a1=a(a+4)a+2

178. Выполните действия:
1) a+2a2−2a+1:a2−43a−3−3a−2;
2) b2+3bb3+9b∗(b−3b+3+b+3b−3);
3) (3c+13c−1−3c−13c+1):2c6c+2;
4) (1a2−4ab+4b2−14b2−a2):2aa2−4b2;
5) (a−8a2−10a+25−aa2−25):a−20(a−5)2;
6) (2x+1x2+6x+9−x−2x2+3x):x2+6x3−9x.

Решение:

1) a+2a2−2a+1:a2−43a−3−3a−2=a+2(a−1)2:(a−2)(a+2)3(a−1)−3a−2=a+2(a−1)2∗3(a−1)(a−2)(a+2)−3a−2=1a−1∗3a−2−3a−2=3(a−1)(a−2)−3a−2=3−3(a−1)(a−1)(a−2)=3−3a+3(a−1)(a−2)=6−3a(a−1)(a−2)=3(2−a)(a−1)(a−2)=−3(a−2)(a−1)(a−2)=−3a−1=31−a

2) b2+3bb3+9b∗(b−3b+3+b+3b−3)=b(b+3)b(b2+9)∗(b−3)2+(b+3)2(b−3)(b+3)=b+3b2+9∗b2−6b+9+b2+6b+9(b−3)(b+3)=1b2+9∗2b2+18b−3=1b2+9∗2(b2+9)b−3=11∗2b−3=2b−3

3) (3c+13c−1−3c−13c+1):2c6c+2=(3c+1)2−(3c−1)2(3c−1)(3c+1):2c2(3c+1)=9c2+6c+1−(9c2−6c+1)(3c−1)(3c+1)∗2(3c+1)2c=12c3c−1∗22c=63c−1∗21=123c−1

4) (1a2−4ab+4b2−14b2−a2):2aa2−4b2=(1(a−2b)2−1(2b−a)(2b+a)):2a(a−2b)(a+2b)=(1(2b−a)2−1(2b−a)(2b+a)):2a(a−2b)(a+2b)=2b+a−(2b−a)(2b−a)2(2b+a)∗(a−2b)(a+2b)2a=2b+a−2b+a(a−2b)2(a+2b)∗(a−2b)(a+2b)2a=2a(a−2b)2(a+2b)∗(a−2b)(a+2b)2a=1a−2b∗11=1a−2b

5) (a−8a2−10a+25−aa2−25):a−20(a−5)2=(a−8(a−5)2−a(a−5)(a+5)):a−20(a−5)2=(a−8)(a+5)−a(a−5)(a−5)2(a+5)∗(a−5)2a−20=a2−8a+5a−40−a2+5aa+5∗1a−20=2a−40a+5∗1a−20=2(a−20)a+5∗1a−20=2a+5

6) (2x+1x2+6x+9−x−2x2+3x):x2+6x3−9x=(2x+1(x+3)2−x−2x(x+3)):x2+6x(x2−9)=x(2x+1)−(x−2)(x+3)x(x+3)2:x2+6x(x−3)(x+3)=2x2+x−(x2−2x+3x−6)x(x+3)2∗x(x−3)(x+3)x2+6=2x2+x−x2+2x−3x+6x(x+3)∗x(x−3)x2+6=x2+6x(x+3)∗x(x−3)x2+6=1x+3∗x−31=x−3x+3

179. Выполните действия:
1) b+4b2−6b+9:b2−162b−6−2b−4;
2) (m−1m+1−m+1m−1):4mm2−1;
3) 2xx2−y2:(1x2+2xy+y2−1y2−x2);
4) (2a−3a2−4a+4−a−1a2−2a):a2−2a3−4a.

Решение:

1) b+4b2−6b+9:b2−162b−6−2b−4=b+4(b−3)2:(b−4)(b+4)2(b−3)−2b−4=b+4(b−3)2∗2(b−3)(b−4)(b+4)−2b−4=1b−3∗2b−4−2b−4=2b−4(1b−3−1)=2b−4∗1−(b−3)b−3=2b−4∗1−b+3b−3=2b−4∗4−bb−3=2b−4∗b−43−b=21∗13−b=23−b

2) (m−1m+1−m+1m−1):4mm2−1=(m−1)2−(m+1)2(m−1)(m+1):4mm2−1=m2−2m+1−(m2+2m+1)m2−1∗m2−14m=m2−2m+1−m2−2m−11∗14m=−4m4m=−1

3) 2xx2−y2:(1x2+2xy+y2−1y2−x2)=2xx2−y2:(1(x+y)2+1x2−y2)=2xx2−y2:(1(x+y)2+1(x−y)(x+y))=2xx2−y2:x−y+x+y(x+y)2(x−y)=2xx2−y2:2x(x+y)2(x−y)=2x(x−y)(x+y)∗(x+y)2(x−y)2x=11∗x+y1=x+y

4) (2a−3a2−4a+4−a−1a2−2a):a2−2a3−4a=(2a−3(a−2)2−a−1a(a−2)):a2−2a(a2−4)=a(2a−3)−(a−1)(a−2)a(a−2)2:a2−2a(a−2)(a+2)=2a2−3a−(a2−a−2a+2)a(a−2)2∗a(a−2)(a+2)a2−2=2a2−3a−a2+a+2a−2a−2∗a+2a2−2=a2−2a−2∗a+2a2−2=1a−2∗a+21=a+2a−2

180. Упростите выражение:
1) (15x−7−x−7)∗7−xx2−16x+64;
2) (a−5a−16a−3):(2a−2aa−3);
3) (1a+2b+ab2)∗aba2−b2+2b−a;
4) (aa−1−aa+1−a2+11−a2):a2+a(a−1)2;
5) (x+2yx−2y−x−2yx+2y−16y2x2−4y2):4yx+2y;
6) (3a−8a2−2a+4+1a+2−4a−28a3+8)∗a2−44.

Решение:

1) (15x−7−x−7)∗7−xx2−16x+64=15−x(x−7)−7(x−7))x−7∗7−x(x−8)2=15−x2+7x−7x+49x−7∗7−x(x−8)2=64−x2x−7∗7−x(x−8)2=x2−647−x∗7−x(x−8)2=(x−8)(x+8)1∗1(x−8)2=x+8x−8

2) (a−5a−16a−3):(2a−2aa−3)=a(a−3)−(5a−16)a−3:2a(a−3)−2aa−3=a2−3a−5a+16a−3:2a2−6a−2aa−3=a2−8a+16a−3:2a2−8aa−3=(a−4)2a−3:2a(a−4)a−3=(a−4)2a−3∗a−32a(a−4)=a−41∗12a=a−42a

3) (1a+2b+ab2)∗aba2−b2+2b−a=b2+2ab+a2ab2∗ab(a−b)(a+b)+2b−a=(a+b)2ab2∗ab(a−b)(a+b)+2b−a=a+bb∗1a−b+2b−a=a+bb(a−b)−2a−b=a+b−2bb(a−b)=a−bb(a−b)=1b

4) (aa−1−aa+1−a2+11−a2):a2+a(a−1)2=(aa−1−aa+1+a2+1a2−1):a2+a(a−1)2=(aa−1−aa+1+a2+1(a−1)(a+1)):a2+a(a−1)2=a(a+1)−a(a−1)+a2+1(a−1)(a+1):a(a+1)(a−1)2=a2+a−a2+a+a2+1(a−1)(a+1):a(a+1)(a−1)2=a2+2a+1(a−1)(a+1):a(a+1)(a−1)2=(a+1)2(a−1)(a+1)∗(a−1)2a(a+1)=11∗a−1a=a−1a

5) (x+2yx−2y−x−2yx+2y−16y2x2−4y2):4yx+2y=(x+2yx−2y−x−2yx+2y−16y2(x−2y)(x+2y)):4yx+2y=(x+2y)2−(x−2y)2−16y2(x−2y)(x+2y)∗x+2y4y=(x+2y−(x−2y))(x+2y+x−2y)−16y2x−2y∗14y=2x(x+2y−x+2y)−16y2x−2y∗14y=2x∗4y−16y2x−2y∗14y=4y(2x−4y)x−2y∗14y=2x−4yx−2y=2(x−2y)x−2y=2

6) (3a−8a2−2a+4+1a+2−4a−28a3+8)∗a2−44=(3a−8a2−2a+4+1a+2−4a−28(a+2)(a2−2a+4))∗a2−44=(3a−8)(a+2)+a2−2a+4−(4a−28)(a+2)(a2−2a+4)∗a2−44=3a2−8a+6a−16+a2−2a+4−4a+28(a+2)(a2−2a+4)∗a2−44=4a2−8a+16(a+2)(a2−2a+4)∗(a−2)(a+2)4=4(a2−2a+4)a2−2a+4∗a−24=11∗a−21=a−2

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