Ответы к странице 140
549. Вынесите множитель из−под знака корня:
1) √18x12√18x12;
2) √y9√y9.
Решение:
1) √18x12=√9∗2∗(x6)2=√9∗√2∗√(x6)2=3√2x6√18x12=√9∗2∗(x6)2=√9∗√2∗√(x6)2=3√2x6
2) √y9=√y8∗y=√(y4)2∗y=√(y4)2∗√y=y4√y√y9=√y8∗y=√(y4)2∗y=√(y4)2∗√y=y4√y
550. Упростите выражение:
1) √98−√50+√32√98−√50+√32;
2) 3√8+√128−13√1623√8+√128−13√162;
3) 0,7√300−7√349+23√1080,7√300−7√349+23√108;
4) √5a−2√20a+3√80a√5a−2√20a+3√80a;
5) √a3b−2a√a5b√a3b−2a√a5b, если a > 0;
6) √c5+4c√c3−5c2√c√c5+4c√c3−5c2√c.
Решение:
1) √98−√50+√32=√49∗2−√25∗2+√16∗2=7√2−5√2+4√2=6√2√98−√50+√32=√49∗2−√25∗2+√16∗2=7√2−5√2+4√2=6√2
2) 3√8+√128−13√162=3√4∗2+√64∗2−13√81∗2=3∗2√2+8√2−13∗9√2=6√2+8√2−3√2=11√23√8+√128−13√162=3√4∗2+√64∗2−13√81∗2=3∗2√2+8√2−13∗9√2=6√2+8√2−3√2=11√2
3) 0,7√300−7√349+23√108=0,7√100∗3−7√149∗3+23√36∗3=0,7∗10√3−7∗17√3+23∗6√3=7√3−√3+2∗2√3=6√3+4√3=10√30,7√300−7√349+23√108=0,7√100∗3−7√149∗3+23√36∗3=0,7∗10√3−7∗17√3+23∗6√3=7√3−√3+2∗2√3=6√3+4√3=10√3
4) √5a−2√20a+3√80a=√5a−2√4∗5a+3√16∗5a=√5a−2∗2√5a+3∗4√5a=√5a−4√5a+12√5a=9√5a√5a−2√20a+3√80a=√5a−2√4∗5a+3√16∗5a=√5a−2∗2√5a+3∗4√5a=√5a−4√5a+12√5a=9√5a
5) √a3b−2a√a5b=√a2∗ab−2a√a4∗ab=a√ab−2a∗a2√ab=a√ab−2a√ab=−a√ab√a3b−2a√a5b=√a2∗ab−2a√a4∗ab=a√ab−2a∗a2√ab=a√ab−2a√ab=−a√ab, если a > 0
6) √c5+4c√c3−5c2√c=√c4∗c+4c√c2∗c−5c2√c=c2√c+4c∗c√c−5c2√c=c2√c+4c2√c−5c2√c=0√c5+4c√c3−5c2√c=√c4∗c+4c√c2∗c−5c2√c=c2√c+4c∗c√c−5c2√c=c2√c+4c2√c−5c2√c=0
551. Упростите выражение:
1) 0,5√12−3√27+0,4√750,5√12−3√27+0,4√75;
2) 2,5√28b+23√63b−10√0,07b2,5√28b+23√63b−10√0,07b;
3) √81a7−5a3√a+6a√a9√81a7−5a3√a+6a√a9.
Решение:
1) 0,5√12−3√27+0,4√75=0,5√4∗3−3√9∗3+0,4√25∗3=0,5∗2√3−3∗3√3+0,4∗5√3=√3−9√3+2√3=−6√30,5√12−3√27+0,4√75=0,5√4∗3−3√9∗3+0,4√25∗3=0,5∗2√3−3∗3√3+0,4∗5√3=√3−9√3+2√3=−6√3
2) 2,5√28b+23√63b−10√0,07b=2,5√4∗7b+23√9∗7b−10√0,01∗7b=2,5∗2√7b+23∗3√7b−10∗0,1√7b=5√7b+2√7b−√7b=6√7b2,5√28b+23√63b−10√0,07b=2,5√4∗7b+23√9∗7b−10√0,01∗7b=2,5∗2√7b+23∗3√7b−10∗0,1√7b=5√7b+2√7b−√7b=6√7b
3) √81a7−5a3√a+6a√a9=√81a6∗a−5a3√a+6a√a8∗a=√81∗(a3)2∗a−5a3√a+6a√(a4)2∗a=9a3√a−5a3√a+6a∗a4√a=4a3√a+6a3√a=10a3√a√81a7−5a3√a+6a√a9=√81a6∗a−5a3√a+6a√a8∗a=√81∗(a3)2∗a−5a3√a+6a√(a4)2∗a=9a3√a−5a3√a+6a∗a4√a=4a3√a+6a3√a=10a3√a
552. Докажите, что:
1) √11+4√7=√7+2√11+4√7=√7+2;
2) √14+8√3=√8+√6√14+8√3=√8+√6.
Решение:
1) √11+4√7=√7+4+4√7=√(√7)2+2∗2√7+22=√(√7+2)2=|√7+2|=√7+2√11+4√7=√7+4+4√7=√(√7)2+2∗2√7+22=√(√7+2)2=|√7+2|=√7+2
2) √14+8√3=√8+6+2∗4√3=√(√8)2+2√16∗3+(√6)2=√(√8)2+2√48+(√6)2=√(√8)2+2√8∗6+(√6)2=√(√8+√6)2=|√8+√6|=√8+√6√14+8√3=√8+6+2∗4√3=√(√8)2+2√16∗3+(√6)2=√(√8)2+2√48+(√6)2=√(√8)2+2√8∗6+(√6)2=√(√8+√6)2=|√8+√6|=√8+√6
553. Упростите выражение:
1) (2√3−1)(√27+2)(2√3−1)(√27+2);
2) (√5−2)2−(3+√5)2;
3) √√17−4∗√√17+4;
4) (7+4√3)(2−√3)2;
5) (√6+2√5−√6−2√5)2.
Решение:
1) (2√3−1)(√27+2)=(2√3−1)(√9∗3+2)=(2√3−1)(3√3+2)=2√3∗3√3+2√3∗2−1∗3√3+(−1)∗2=6∗3+4√3−3√3−2=18+√3−2=16+√3
2) (√5−2)2−(3+√5)2=(√5)2−2∗2√5+22)−(32+3∗2√5+(√5)2)=5−4√5+4−(9+6√5+5)=9−4√5−(14+6√5)=9−4√5−14−6√5=−5−10√5
3) √√17−4∗√√17+4=√(√17−4)((√17+4))=√(√17)2−42=√17−16=√1=1
4) (7+4√3)(2−√3)2=(7+4√3)(22−2∗2√3+(√3)2)=(7+4√3)(4−4√3+3)=(7+4√3)(7−4√3)=72−(4√3)2=49−16∗3=49−48=1
5) (√6+2√5−√6−2√5)2=(√6+2√5)2−2√6+2√5√6−2√5+(√6−2√5)2=6+2√5−2√(6−2√5)(6+2√5)+6−2√5=12−2√62−(2√5)2=12−2√36−4∗5=12−2√36−20=12−2√16=12−2∗4=12−8=4
554. Найдите значение выражения:
1) (3√2+1)(√8−2);
2) (3−2√7)2+(3+2√7)2;
3) (10−4√6)(2+√6)2;
4) (√9−4√2+√9+4√2)2.
Решение:
1) (3√2+1)(√8−2)=(3√2+1)(√4∗2−2)=(3√2+1)(2√2−2)=3√2∗2√2−3√2∗2+1∗2√2−1∗2=6∗2−6√2+2√2−2=12−4√2−2=10−4√2
2) (3−2√7)2+(3+2√7)2=32−2∗3∗2√7+(2√7)2+32+2∗3∗2√7+(2√7)2=9−12√7+4∗7+9+12√7+4∗7=18+28+28=74
3) (10−4√6)(2+√6)2=(10−4√6)(22+2∗2√6+(√6)2)=(10−4√6)(4+4√6+6)=(10−4√6)(10+4√6)=102−(4√6)2=100−16∗6=100−96=4
4) (√9−4√2+√9+4√2)2=(√9−4√2)2+2√9−4√2√9+4√2+(√9+4√2)2=9−4√2+2√(9−4√2)(9+4√2)+9+4√2=18+2√92−(4√2)2=18+2√81−16∗2=18+2√81−32=18+2√49=18+2∗7=18+14=32
555. Сократите дробь:
1) 4a+4√5a2−5;
2) √28−2√2a6a−21;
3) a+4√ab+4ba−4b, если a > 0, b > 0;
4) x2−6yx2+6y−x√24y;
5) √a+√b√a3+√b3;
6) m√m−27√m−3.
Решение:
1) 4a+4√5a2−5=4(a+√5)(a−√5)(a+√5)=4a−√5
2) √28−2√2a6a−21=√4∗7−2√2a3(2a−7)=2√7−2√2a3((√2a)2−(√7)2)=2(√7−√2a)3(√2a−√7)(√2a+√7)=−2(√2a−√7)3(√2a−√7)(√2a+√7)=−23(√2a+√7)
3) a+4√ab+4ba−4b=(√a)2+2∗√a∗2√b+(2√b)2(√a)2−(2√b)2=(√a+2√b)2(√a−2√b)(√a+2√b)=√a+2√b√a−2√b, если a > 0, b > 0
4) x2−6yx2+6y−x√24y=x2−(√6y)2x2−x√4∗6y+(√6y)2=(x−√6y)(x+√6y)x2−2x√6y+(√6y)2=(x−√6y)(x+√6y)(x−√6y)2=x+√6yx−√6y
5) √a+√b√a3+√b3=√a+√b(√a)3+(√b)3=√a+√b(√a+√b)((√a)2−√a√b+(√b)2)=√a+√b(√a+√b)(a−√a√b+b)=1a−√a√b+b
6) m√m−27√m−3=√m2∗m−27√m−3=√m3−27√m−3=(√m)3−33√m−3=(√m−3)((√m)2)+3√m+32)√m−3=m+3√m+9
556. Сократите дробь:
1) a−b√11b−√11a;
2) 2a+10√2ab+25b6a−75b, если a > 0, b > 0;
3) a−2√a+4a√a+8.
Решение:
1) a−b√11b−√11a=(√a)2−(√b)2√11∗√b−√11∗√a=(√a−√b)(√a+√b)√11(√b−√a)=−(√a−√b)(√a+√b)√11(√a−√b)=−√a+√b√11
2) 2a+10√2ab+25b6a−75b=(√2a)2+10√2ab+(√25b)23(2a−25b)=(√2a)2+2∗√2a∗5√b+(5√b)23((√2a)2−(√25b)2)=(√2a+5√b)23((√2a)2−(5√b)2)=(√2a+5√b)23(√2a−5√b)(√2a+5√b)=√2a+5√b3(√2a−5√b), если a > 0, b > 0
3) a−2√a+4a√a+8=a−2√a+4√a2∗a+8=a−2√a+4√a3+8=a−2√a+4(√a)3+23=a−2√a+4(√a+2)((√a)2−2√a+22)=a−2√a+4(√a+2)(a−2√a+4)=1√a+2
