1 вариант:
а)
![5^1*5^{-5}=5^{1+(-5)}=5^{-4}=\frac{1}{5^4}=\frac{1}{625}](https://tex.z-dn.net/?f=5%5E1%2A5%5E%7B-5%7D%3D5%5E%7B1%2B%28-5%29%7D%3D5%5E%7B-4%7D%3D%5Cfrac%7B1%7D%7B5%5E4%7D%3D%5Cfrac%7B1%7D%7B625%7D)
б)
![3*(\frac{1}{3})^{-2}=3^1*3^{-1*-2}=3^1*3^2=3^{1+2}=3^3=27](https://tex.z-dn.net/?f=3%2A%28%5Cfrac%7B1%7D%7B3%7D%29%5E%7B-2%7D%3D3%5E1%2A3%5E%7B-1%2A-2%7D%3D3%5E1%2A3%5E2%3D3%5E%7B1%2B2%7D%3D3%5E3%3D27)
в)
![\frac{4^{-3}*2^6}{8}=\frac{2^{2*-3}*2^6}{2^3}=2^{-6+6-3}=2^{-3}=\frac{1}{2^3}=\frac{1}{8}](https://tex.z-dn.net/?f=%5Cfrac%7B4%5E%7B-3%7D%2A2%5E6%7D%7B8%7D%3D%5Cfrac%7B2%5E%7B2%2A-3%7D%2A2%5E6%7D%7B2%5E3%7D%3D2%5E%7B-6%2B6-3%7D%3D2%5E%7B-3%7D%3D%5Cfrac%7B1%7D%7B2%5E3%7D%3D%5Cfrac%7B1%7D%7B8%7D)
2 вариант.
а)
![2^1*2^{-3}=2^{1-3}=2^{-2}=\frac{1}{4}](https://tex.z-dn.net/?f=2%5E1%2A2%5E%7B-3%7D%3D2%5E%7B1-3%7D%3D2%5E%7B-2%7D%3D%5Cfrac%7B1%7D%7B4%7D)
б)
![(\frac{1}{4})^{-2}*4=4^{-1*-2}*4^1=4^{2+1}=4^3=64](https://tex.z-dn.net/?f=%28%5Cfrac%7B1%7D%7B4%7D%29%5E%7B-2%7D%2A4%3D4%5E%7B-1%2A-2%7D%2A4%5E1%3D4%5E%7B2%2B1%7D%3D4%5E3%3D64)
в)
![\frac{((3)^{-2})^3*27^2}{3}=\frac{3^{-2*3}*3^{3*2}}{3}=\frac{3^{-6+6}}{3}=\frac{1}{3}](https://tex.z-dn.net/?f=%5Cfrac%7B%28%283%29%5E%7B-2%7D%29%5E3%2A27%5E2%7D%7B3%7D%3D%5Cfrac%7B3%5E%7B-2%2A3%7D%2A3%5E%7B3%2A2%7D%7D%7B3%7D%3D%5Cfrac%7B3%5E%7B-6%2B6%7D%7D%7B3%7D%3D%5Cfrac%7B1%7D%7B3%7D)
Качество фото не лучшее, поэтому мог записать некоторые примеры неправильно.
Через 550 прыжков он окажется правее точки 1100.
A²(1-a)+4(a-1)=a²(1-a)-4(1-a)=(1-a)(a²-4)=(1-a)(a-2)(a+2)
![\dfrac{\log_4(2^x-1)}{x-1}\leq 1](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Clog_4%282%5Ex-1%29%7D%7Bx-1%7D%5Cleq+1)
Область допустимых значений
![\displaystyle \left \{ {{2^x-1>0} \atop {x-1\neq 0}} \right. ~~\Leftrightarrow~~ \left \{ {{2^x>2^0} \atop {x\neq 1}} \right. ~~\Leftrightarrow~~ \left \{ {{x>0} \atop {x\neq 1}} \right.](https://tex.z-dn.net/?f=%5Cdisplaystyle+%5Cleft+%5C%7B+%7B%7B2%5Ex-1%3E0%7D+%5Catop+%7Bx-1%5Cneq+0%7D%7D+%5Cright.+~~%5CLeftrightarrow~~+%5Cleft+%5C%7B+%7B%7B2%5Ex%3E2%5E0%7D+%5Catop+%7Bx%5Cneq+1%7D%7D+%5Cright.+~~%5CLeftrightarrow~~+%5Cleft+%5C%7B+%7B%7Bx%3E0%7D+%5Catop+%7Bx%5Cneq+1%7D%7D+%5Cright.)
<em>ОДЗ : x ∈ (0; 1) ∪ (1; +∞)</em>
![1)~x\in (0;1); ~~(x-1)<0~~ \Rightarrow\\\\\dfrac{\log_4(2^x-1)}{x-1}\leq 1~~\Big|\cdot (x-1)<0\\\\\log_4(2^x-1)\geq x-1~~\Leftrightarrow~~\log_4(2^x-1)\geq \log_44^{x-1}\\\\2^x-1\geq 4^{x-1}~~|\cdot4~~~~\Leftrightarrow~~4\cdot 2^x-4\geq 4^x\\\\2^{2x}-4\cdot 2^x+4\leq 0\\\\(2^x-2)^2\leq 0;~~~2^x=2;~~~x=1;](https://tex.z-dn.net/?f=1%29~x%5Cin+%280%3B1%29%3B+~~%28x-1%29%3C0~~+%5CRightarrow%5C%5C%5C%5C%5Cdfrac%7B%5Clog_4%282%5Ex-1%29%7D%7Bx-1%7D%5Cleq+1~~%5CBig%7C%5Ccdot+%28x-1%29%3C0%5C%5C%5C%5C%5Clog_4%282%5Ex-1%29%5Cgeq+x-1~~%5CLeftrightarrow~~%5Clog_4%282%5Ex-1%29%5Cgeq+%5Clog_44%5E%7Bx-1%7D%5C%5C%5C%5C2%5Ex-1%5Cgeq+4%5E%7Bx-1%7D~~%7C%5Ccdot4~~~~%5CLeftrightarrow~~4%5Ccdot+2%5Ex-4%5Cgeq+4%5Ex%5C%5C%5C%5C2%5E%7B2x%7D-4%5Ccdot+2%5Ex%2B4%5Cleq+0%5C%5C%5C%5C%282%5Ex-2%29%5E2%5Cleq+0%3B~~~2%5Ex%3D2%3B~~~x%3D1%3B)
Не подходит по ОДЗ
![2)~x\in (1;+\infty ); ~~(x-1)>0~~ \Rightarrow\\\\\dfrac{\log_4(2^x-1)}{x-1}\leq 1~~\Big|\cdot (x-1)>0\\\\\log_4(2^x-1)\leq x-1~~\Leftrightarrow~~\log_4(2^x-1)\leq \log_44^{x-1}\\\\2^x-1\leq 4^{x-1}~~|\cdot4~~~~\Leftrightarrow~~4\cdot 2^x-4\leq 4^x\\\\2^{2x}-4\cdot 2^x+4\geq 0\\\\(2^x-2)^2\geq 0;](https://tex.z-dn.net/?f=2%29~x%5Cin+%281%3B%2B%5Cinfty+%29%3B+~~%28x-1%29%3E0~~+%5CRightarrow%5C%5C%5C%5C%5Cdfrac%7B%5Clog_4%282%5Ex-1%29%7D%7Bx-1%7D%5Cleq+1~~%5CBig%7C%5Ccdot+%28x-1%29%3E0%5C%5C%5C%5C%5Clog_4%282%5Ex-1%29%5Cleq+x-1~~%5CLeftrightarrow~~%5Clog_4%282%5Ex-1%29%5Cleq+%5Clog_44%5E%7Bx-1%7D%5C%5C%5C%5C2%5Ex-1%5Cleq+4%5E%7Bx-1%7D~~%7C%5Ccdot4~~~~%5CLeftrightarrow~~4%5Ccdot+2%5Ex-4%5Cleq+4%5Ex%5C%5C%5C%5C2%5E%7B2x%7D-4%5Ccdot+2%5Ex%2B4%5Cgeq+0%5C%5C%5C%5C%282%5Ex-2%29%5E2%5Cgeq+0%3B)
Квадрат выражения всегда неотрицательный.
<em>Ответ: х ∈ (1; +∞)</em>
1.
1) 1000m^3-n^3=(10m)^3-n^3= (10m-n)(100m^2+10mn+n^2)
2) 81a^3-ab^2=a(81a^2-b^2)=a(9a-b)(9a+b)
3) -8x^2-16xy-8y^2= -8(x^2+2xy+y^2)=-2(x+y)^2= -8(x+y)(x+y)
4) 5mn-10n+15m-30=5m(x+3)-10n(n+3)=(n+3)(5m-10n)=5(n+3)(m-2n)
3.
1) a^2-36b^2+a-6b=(a-6b)(a+6b) +(a-6b)= (a-6b)(a+6b+1)
2)25x^2-10xy+y^2-9=(5x-y)^2-9=(5x-y-3)(5x-y+3)
3) ay^7+y^7-ay^3-y^3=y^7(a+1)-y^3(a+1)=(a+1)(y^7-y^3)
4) 4-m^2+14mn-49n^2=4-(m^2-14mn+49m^2)=4-(m-7n)^2=
= [2-(m-7n)][ 2+(m-7n)]=(2-m-7n)(2+m-7n)
4.
1) 2x^2-32x=0
2x(x^2-16)=0
2x=0 , x-4=0 , x+4=0
X1=0 x2=4 x3=-4
2) 81x^3+18x^2+x=0
X(81x^2+18x+1)=0
X(x+1/9)(x+1/9)=0
X1=0 , x+1/9=0
X2=-1/9
3) x^3+6x^2-x-6=0
X^2(x+6)-(x+6)=0
(x+6)(x^2-1)=0
(x+6)(x-1)(x+1)=0
X=6 , x-1=0 , x+1=0
X1=-6 x2=1 x3=-1
³