\(7^6+7^5-7^4=7^4\left(7^2+7-1\right)=7^4.55=7^3.7.11.5=5.77.7^3\)

Do 77 chia hết 77 \(\Rightarrow5.77.7^3⋮77\)

Vậy \(\left(7^6+7^5-7^4\right)⋮77\)

a.

\(\Leftrightarrow\sqrt{2x^2+5x+3}=-x-3\)

\(\Leftrightarrow\left\{{}\begin{matrix}-x-3\ge0\\2x^2+5x+3=\left(-x-3\right)^2\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}x\le-3\\2x^2+5x+3=x^2+6x+9\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}x\le-3\\x^2-x-6=0\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}x\le-3\\\left[{}\begin{matrix}x=3\left(loại\right)\\x=-2\left(loại\right)\end{matrix}\right.\end{matrix}\right.\)

Vậy pt vô nghiệm

b.

\(\Leftrightarrow\sqrt{2x^2+x+3}=1-x\)

\(\Leftrightarrow\left\{{}\begin{matrix}1-x\ge0\\2x^2+x+3=\left(1-x\right)^2\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}x\le1\\2x^2+x+3=1-2x+x^2\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}x\le1\\x^2+3x+2=0\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}x\le1\\\left[{}\begin{matrix}x=-1\left(loại\right)\\x=-2\left(loại\right)\end{matrix}\right.\end{matrix}\right.\)

Vậy pt vô nghiệm

\(\Leftrightarrow\left(x+5\right)^2-9\left(x+5\right)=0\)

\(\Leftrightarrow\left(x+5\right)\left(x+5-9\right)=0\)

\(\Leftrightarrow\left(x+5\right)\left(x-4\right)=0\)

\(\Rightarrow\left[{}\begin{matrix}x=-5\\x=4\end{matrix}\right.\)

\(\left(x+5\right)^2-9x-45x=0\\ < =>\left(x^2+10x+25\right)-54x=0\\ < =>x^2+10x+25-54x=0\\ < =>x^2-44x+25=0\\ < =>\left(x^2-44x+484\right)-459=0\\ < =>\left(x-22\right)^2-459=0\\ < =>\left(x-22\right)^2=459\\ < =>\left[{}\begin{matrix}x-22=\sqrt{459}\\x-22=-\sqrt{459}\end{matrix}\right.\\ < =>\left[{}\begin{matrix}x=22+\sqrt{459}\\x=22-\sqrt{459}\end{matrix}\right.\)

a: Xét ΔMNP có \(\dfrac{MH}{MN}=\dfrac{MK}{MP}\)

nên HK//PN

Xét tứ giác NHKP có HK//NP

nên NHKP là hình thang

Hình thang NHKP có \(\widehat{HNP}=\widehat{KPN}\)(ΔMNP cân tại M)

nên NHKP là hình thang cân

`2x + 3x + 1 - 4x + 2 = 36`

`=> (2+3-4) x + 3 = 36`

`=> x + 3 = 36`

`=> x = 36-3`

`=> x = 33`

Tìm x biết.                                                180-(x_45):2=120

\(5^7-5^6+5^5\\ =5^5\cdot5^2-5^5\cdot5+5^5\\ =5^5\cdot\left(5^2-5+1\right)\\ =5^5\cdot\left(25-5+1\right)\\ =5^5\cdot21⋮21\)

=> `5^7-5^6+5^5` chia hết cho 21 

`5^7 - 5^6 + 5^5`

`= 5^5 . (5^2 - 5 + 1)`

`= 5^5 . (25 - 5 + 1)`

`= 5^5 . 21 vdots 21 (đpcm)`

a: Thay m=3 vào hệ, ta được:

\(\left\{{}\begin{matrix}3x+2y=1\\3x+\left(3+1\right)y=-1\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}3x+2y=1\\3x+4y=-1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}3x+4y-3x-2y=-1-1\\3x+2y=1\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}2y=-2\\3x=1-2y=1-\left(-2\right)=3\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}y=-1\\x=1\end{matrix}\right.\)

b:

để hệ có vô số nghiệm thì \(\dfrac{m}{3}=\dfrac{2}{m+1}=\dfrac{1}{-1}\)

=>\(\left\{{}\begin{matrix}m^2+m=6\\m+1=-2\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}\left(m+3\right)\left(m-2\right)=0\\m=-3\end{matrix}\right.\)

=>m=-3

Để hệ vô nghiệm thì \(\dfrac{m}{3}=\dfrac{2}{m+1}\ne\dfrac{1}{-1}=-1\)

=>\(\left\{{}\begin{matrix}m^2+m=6\\m+1\ne-2\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}\left(m+3\right)\left(m-2\right)=0\\m\ne-3\end{matrix}\right.\)

=>m=2

Để hệ có nghiệm duy nhất thì \(\dfrac{m}{3}\ne\dfrac{2}{m+1}\)

=>\(m^2+m\ne6\)

=>\(m^2+m-6\ne0\)

=>(m+3)(m-2)<>0

=>\(m\notin\left\{-3;2\right\}\)

\(\left\{{}\begin{matrix}mx+2y=1\\3x+\left(m+1\right)y=-1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}3mx+6y=3\\3mx+\left(m^2+m\right)y=-m\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}3mx+\left(m^2+m\right)y-3mx-6y=-m-3\\mx+2y=1\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}y\left(m+3\right)\left(m-2\right)=-\left(m+3\right)\\mx+2y=1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}y=\dfrac{-1}{m-2}\\mx=1-2y=1+\dfrac{2}{m-2}=\dfrac{m-2+2}{m-2}=\dfrac{m}{m-2}\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}y=-\dfrac{1}{m-2}\\x=\dfrac{1}{m-2}\end{matrix}\right.\)

c: Để hệ có nghiệm duy nhất là số nguyên thì \(\left\{{}\begin{matrix}m\in\left\{-3;2\right\}\\m-2\inƯC\left(1;-1\right)\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}m\in\left\{-3;2\right\}\\m-2\in\left\{1;-1\right\}\end{matrix}\right.\)

=>\(m\in\left\{3;1\right\}\)

Câu hỏi là gì vậy bạn?

Do 8 chia hết cho 4 \(\Rightarrow8^{2008}⋮4\)

\(\Rightarrow8^{2008}=4k\)

\(\Rightarrow5^{8^{2008}}=5^{4k}=\left(5^4\right)^k=625^k\)

Mà \(625\equiv1\left(mod24\right)\Rightarrow625^k\equiv1\left(mod24\right)\)

\(\Rightarrow5^{8^{2008}}\equiv1\left(mod24\right)\)

\(\Rightarrow5^{8^{2008}}+23\equiv0\left(mod24\right)\)

Hay \(5^{8^{2008}}+23\) chia hết 24

(x - 45) x 27 = 0

=> x - 45 = 0 : 27 

=> x - 45 = 0 

=> x = 0 + 45

=> x = 45 

Vậy: .. 

`(x - 45) . 27 = 0`

`=> x - 45 = 0 : 27`

`=> x - 45 = 0`

`=> x = 45`

Vậy `x = 45`