Упр.1113 ГДЗ Алимов 10-11 класс (Алгебра)

1) (3a+1)5;
2) (x+6)6;
3) (x+1/3)7;
4) (a/3-1)5;
5) (10x-0,1)6;
6) (0,1b-10)7;
7) (2/a + a/2)7;
8) (2/c + c/2).

Используем формулу бинома Ньютона:

$$ (a+b)^n=\sum_{k=0}^{n} C_n^k a^{n-k}b^k $$

1. $$ (3a+1)^5=(3a)^5+5(3a)^4+10(3a)^3+10(3a)^2+5(3a)+1 $$

   $$ =243a^5+405a^4+270a^3+90a^2+15a+1 $$

2. $$ (x+6)^6=x^6+6x^5\cdot 6+15x^4\cdot 6^2+20x^3\cdot 6^3+15x^2\cdot 6^4+6x\cdot 6^5+6^6 $$

   $$ =x^6+36x^5+540x^4+4320x^3+19440x^2+46656x+46656 $$

3. $$ \left(x+\frac13\right)^7=x^7+7x^6\cdot \frac13+21x^5\cdot \frac1{3^2}+35x^4\cdot \frac1{3^3}+35x^3\cdot \frac1{3^4}+21x^2\cdot \frac1{3^5}+7x\cdot \frac1{3^6}+\frac1{3^7} $$

   $$ =x^7+\frac73x^6+\frac73x^5+\frac{35}{27}x^4+\frac{35}{81}x^3+\frac7{243}x^2+\frac7{729}x+\frac1{2187} $$

4. $$ \left(\frac a3-1\right)^5=\left(\frac a3\right)^5-5\left(\frac a3\right)^4+10\left(\frac a3\right)^3-10\left(\frac a3\right)^2+5\left(\frac a3\right)-1 $$

   $$ =\frac{1}{243}a^5-\frac{5}{81}a^4+\frac{10}{27}a^3-\frac{10}{9}a^2+\frac53a-1 $$

5. $$ (10x-0{,}1)^6=(10x)^6-6(10x)^5\cdot 0{,}1+15(10x)^4\cdot 0{,}1^2-20(10x)^3\cdot 0{,}1^3+15(10x)^2\cdot 0{,}1^4-6(10x)\cdot 0{,}1^5+0{,}1^6 $$

   $$ =1000000x^6-600000x^5+150000x^4-20000x^3+1500x^2-60x+0{,}000001 $$

6. $$ (0{,}1b-10)^7=\left(\frac b{10}-10\right)^7 $$

   $$ =\left(\frac b{10}\right)^7-7\left(\frac b{10}\right)^6\cdot 10+21\left(\frac b{10}\right)^5\cdot 10^2-35\left(\frac b{10}\right)^4\cdot 10^3+35\left(\frac b{10}\right)^3\cdot 10^4-21\left(\frac b{10}\right)^2\cdot 10^5+7\left(\frac b{10}\right)\cdot 10^6-10^7 $$

   $$ =\frac{b^7}{10^7}-\frac{7b^6}{10^5}+\frac{21b^5}{10^3}-35b^4+350b^3-21000b^2+700000b-10000000 $$

7. $$ \left(\frac2a+\frac a2\right)^7=\left(\frac2a\right)^7+7\left(\frac2a\right)^6\cdot \frac a2+21\left(\frac2a\right)^5\cdot \left(\frac a2\right)^2+35\left(\frac2a\right)^4\cdot \left(\frac a2\right)^3+35\left(\frac2a\right)^3\cdot \left(\frac a2\right)^4+21\left(\frac2a\right)^2\cdot \left(\frac a2\right)^5+7\left(\frac2a\right)\cdot \left(\frac a2\right)^6+\left(\frac a2\right)^7 $$

   $$ =\frac{128}{a^7}+\frac{224}{a^5}+\frac{168}{a^3}+\frac{70}{a}+\frac{35}{2}a+\frac{21}{8}a^3+\frac{7}{32}a^5+\frac{1}{128}a^7 $$

8. $$ \left(\frac2c+\frac c2\right)^8=\left(\frac2c\right)^8+8\left(\frac2c\right)^7\cdot \frac c2+28\left(\frac2c\right)^6\cdot \left(\frac c2\right)^2+56\left(\frac2c\right)^5\cdot \left(\frac c2\right)^3+70\left(\frac2c\right)^4\cdot \left(\frac c2\right)^4+56\left(\frac2c\right)^3\cdot \left(\frac c2\right)^5+28\left(\frac2c\right)^2\cdot \left(\frac c2\right)^6+8\left(\frac2c\right)\cdot \left(\frac c2\right)^7+\left(\frac c2\right)^8 $$

   $$ =\frac{256}{c^8}+\frac{512}{c^6}+\frac{448}{c^4}+\frac{224}{c^2}+70+14c^2+\frac74c^4+\frac18c^6+\frac1{256}c^8 $$

Ответ

1. $$243a^5+405a^4+270a^3+90a^2+15a+1$$
2. $$x^6+36x^5+540x^4+4320x^3+19440x^2+46656x+46656$$
3. $$x^7+\frac73x^6+\frac73x^5+\frac{35}{27}x^4+\frac{35}{81}x^3+\frac7{243}x^2+\frac7{729}x+\frac1{2187}$$
4. $$\frac{1}{243}a^5-\frac{5}{81}a^4+\frac{10}{27}a^3-\frac{10}{9}a^2+\frac53a-1$$
5. $$1000000x^6-600000x^5+150000x^4-20000x^3+1500x^2-60x+0{,}000001$$
6. $$\frac{b^7}{10^7}-\frac{7b^6}{10^5}+\frac{21b^5}{10^3}-35b^4+350b^3-21000b^2+700000b-10000000$$
7. $$\frac{128}{a^7}+\frac{224}{a^5}+\frac{168}{a^3}+\frac{70}{a}+\frac{35}{2}a+\frac{21}{8}a^3+\frac{7}{32}a^5+\frac{1}{128}a^7$$
8. $$\frac{256}{c^8}+\frac{512}{c^6}+\frac{448}{c^4}+\frac{224}{c^2}+70+14c^2+\frac74c^4+\frac18c^6+\frac1{256}c^8$$