Упр.373 ГДЗ Муравин 7 класс (Алгебра)

1. Вычислите с помощью формул сокращённого умножения: 1) $$(100+1)^2$$; 2) $$(80-1)^2$$; 3) $$10{,}01^2$$; 4) $$9{,}98^2$$; 5) $$99\cdot101$$; 6) $$201\cdot199$$; 7) $$126^2-74^2$$; 8) $$356^2-144^2$$; 9) $$37^2+2\cdot37\cdot63+63^2$$; 10) $$83^2+33^2-83\cdot66$$; 11) $$19{,}3^2+2\cdot19{,}3\cdot30{,}7+30{,}7^2$$; 12) $$31{,}8^2-2\cdot31{,}8\cdot21{,}8+21{,}8^2$$.

1. $$\begin{aligned} (100+1)^2&=100^2+2\cdot100\cdot1+1^2\\ &=10000+200+1=10201 \end{aligned}$$
2. $$\begin{aligned} (80-1)^2&=80^2-2\cdot80\cdot1+1^2\\ &=6400-160+1=6241 \end{aligned}$$
3. $$\begin{aligned} 10{,}01^2&=(10+0{,}01)^2\\ &=10^2+2\cdot10\cdot0{,}01+0{,}01^2\\ &=100+0{,}2+0{,}0001=100{,}2001 \end{aligned}$$
4. $$\begin{aligned} 9{,}98^2&=(10-0{,}02)^2\\ &=10^2-2\cdot10\cdot0{,}02+0{,}02^2\\ &=100-0{,}4+0{,}0004=99{,}6004 \end{aligned}$$
5. $$\begin{aligned} 99\cdot101&=(100-1)(100+1)\\ &=100^2-1^2=10000-1=9999 \end{aligned}$$
6. $$\begin{aligned} 201\cdot199&=(200+1)(200-1)\\ &=200^2-1^2=40000-1=39999 \end{aligned}$$
7. $$\begin{aligned} 126^2-74^2&=(126-74)(126+74)\\ &=52\cdot200=10400 \end{aligned}$$
8. $$\begin{aligned} 356^2-144^2&=(356-144)(356+144)\\ &=212\cdot500=106000 \end{aligned}$$
9. $$\begin{aligned} 37^2+2\cdot37\cdot63+63^2&=(37+63)^2\\ &=100^2=10000 \end{aligned}$$
10. $$\begin{aligned} 83^2+33^2-83\cdot66&=83^2-2\cdot83\cdot33+33^2\\ &=(83-33)^2=50^2=2500 \end{aligned}$$
11. $$\begin{aligned} 19{,}3^2+2\cdot19{,}3\cdot30{,}7+30{,}7^2&=(19{,}3+30{,}7)^2\\ &=50^2=2500 \end{aligned}$$
12. $$\begin{aligned} 31{,}8^2-2\cdot31{,}8\cdot21{,}8+21{,}8^2&=(31{,}8-21{,}8)^2\\ &=10^2=100 \end{aligned}$$

Ответ

1) 10201; 2) 6241; 3) 100,2001; 4) 99,6004; 5) 9999; 6) 39999; 7) 10400; 8) 106000; 9) 10000; 10) 2500; 11) 2500; 12) 100.