\(a.5\cdot3^x=5\cdot3^4\\ =>3^x=\dfrac{5\cdot3^4}{5}=3^4\\ =>x=4\\ b.7\cdot4^x=7\cdot4^3\\ =>4^x=\dfrac{7\cdot4^3}{7}=4^3\\ =>x=3\\ c.\dfrac{3}{5}\cdot4^x=7\cdot4^3\\ =>4^x=\dfrac{7\cdot4^3}{\dfrac{3}{5}}=\dfrac{35}{3}\cdot4^3\\ =>\dfrac{4^x}{4^3}=\dfrac{35}{3}\\ =>4^{x-3}=\dfrac{35}{3}\\ =>x-3=log_4\dfrac{35}{3}\\ =>x=log_4\dfrac{35}{3}+3\\ d.\dfrac{3}{2}\cdot5^x=\dfrac{3}{2}\cdot5^{12}\\ =>5^x=\dfrac{5^{12}\cdot\dfrac{3}{2}}{\dfrac{3}{2}}=5^{12}\\ =>x=12\) 

e: \(9\cdot5^x=6\cdot5^6+3\cdot5^6\)

=>\(9\cdot5^x=9\cdot5^6\)

=>\(5^x=5^6\)

=>x=6

f: \(5\cdot3^x=7\cdot3^5-2\cdot3^5\)

=>\(5\cdot3^x=5\cdot3^5\)

=>\(3^x=3^5\)

=>x=5

g: \(5\cdot3^{x+6}=2\cdot3^5+3\cdot3^5\)

=>\(5\cdot3^{x+6}=5\cdot3^5\)

=>\(3^{x+6}=3^5\)

=>x+6=5

=>x=-1

\(x:0,25+x\times11=24\)

\(x\times4+x\times11=24\)

\(x\times\left(4+11\right)=24\)

\(x\times15=24\)

\(x=24:15\)

\(x=1,6\)

\(\left(x+1\right)^2-\left(x-1\right)^2\\ =\left[\left(x+1\right)-\left(x-1\right)\right]\left[\left(x+1\right)+\left(x-1\right)\right]\\ =\left(x+1-x+1\right)\left(x+1+x-1\right)\\ =2\cdot2x\\ =4x\)

\(a.\left(x+y+4\right)\left(x+y-4\right)\\ =\left[\left(x+y\right)+4\right]\left[\left(x+y\right)-4\right]\\ =\left(x+y\right)^2-4^2\\ b.\left(x-y+6\right)\left(x+y-6\right)\\ =\left[x-\left(y-6\right)\right]\left[x+\left(y-6\right)\right]\\ =x^2-\left(y-6\right)^2\\ c.\left(y+2z-3\right)\left(y-2z-3\right)\\ =\left[\left(y-3\right)+2z\right]\left[\left(y-3\right)-2z\right]\\ =\left(y-3\right)^2-\left(2z\right)^2\\ d.\left(x+2y+3z\right)\left(2y+3z-x\right)\\ =\left[\left(2y+3z\right)+x\right]\left[\left(2y+3z\right)-x\right]\\ =\left(2y+3z\right)^2-x^2\)

\(a.\left(x+1\right)^2-\left(x-1\right)^2-3\left(x+1\right)\left(x-1\right)\\ =\left(x^2+2x+1\right)-\left(x^2-2x+1\right)-3\left(x^2-1\right)\\ =x^2+2x+1-x^2+2x-1-3x^2+3\\ =4x-3x^2+3\\b.5\left(x-2\right)\left(x+2\right)-\dfrac{1}{2}\left(6-8x\right)^2+17\\ =5\left(x^2-4\right)-\dfrac{1}{2}\left(36-96x+64x^2\right)+17\\ =5x^2-20-18+48x-32x^2\\ =48x-27x^2-38\)

\(a.A=9x^2+42x+49\\ =\left(3x\right)^2+2\cdot3x\cdot7+7^2\\ =\left(3x+7\right)^2\)

Thay x = 1 vào A ta có:

`A=(3*1+7)^2=10^2=100` 

\(b.B=25x^2-2xy+\dfrac{1}{25}y^2\\ =\left(5x\right)^2-2\cdot5x\cdot\dfrac{1}{5}y+\left(\dfrac{1}{5}y\right)^2\\ =\left(5x-\dfrac{1}{5}y\right)^2\)

Thay x = -1/5 và y = -5 vào B ta có:

\(B=\left(5\cdot\dfrac{-1}{5}-\dfrac{1}{5}\cdot-5\right)^2=\left(-1+1\right)^2=0\)

Có a, b ϵ N

Ta có:

a : 24 = b (dư 10)

a = b x 24 + 10

Do cả 24 và 10 đều chia hết cho 2 ⇒ a ⋮ 2

Nhưng chỉ có 24 ⋮ 4 còn 10 thì không nên ⇒ a \(⋮̸\)4

\(a.25x^2-9=0\\ \Leftrightarrow\left(5x\right)^2-3^2=0\\ \Leftrightarrow\left(5x+3\right)\left(5x-3\right)=0\\ \Leftrightarrow\left[{}\begin{matrix}5x=3\\5x=-3\end{matrix}\right.\\ \Leftrightarrow\left[{}\begin{matrix}x=\dfrac{3}{5}\\x=-\dfrac{3}{5}\end{matrix}\right.\\ b.\left(x+4\right)^2-\left(x+1\right)\left(x-1\right)=16\\ \Leftrightarrow x^2+8x+16-x^2+1=16\\ \Leftrightarrow8x+17=16\\ \Leftrightarrow8x=-1\\ \Leftrightarrow x=-\dfrac{1}{8}\\ c.\left(2x-1\right)^2+\left(x+3\right)^2-5\left(x+7\right)\left(x-7\right)=0\\ \Leftrightarrow4x^2-4x+1+x^2+6x+9-5\left(x^2-49\right)=0\\ \Leftrightarrow5x^2+2x+10-5x^2+245=0\\ \Leftrightarrow2x+265=0\\ \Leftrightarrow2x=-265\\ \Leftrightarrow x=-\dfrac{265}{2}\)

\(a.A=x^2+5x+7\\ =\left[x^2+2\cdot x\cdot\dfrac{5}{2}+\left(\dfrac{5}{2}\right)^2\right]+\dfrac{3}{4}\\ =\left(x+\dfrac{5}{2}\right)^2+\dfrac{3}{4}\ge\dfrac{3}{4}\forall x\)

Dấu "=" xảy ra: `x+5/2=0<=>x=-5/2` 

\(b.B=6x-x^2-5\\ =-\left(x^2-6x+9\right)+4\\ =-\left(x-3\right)^2+4\le4\forall x\)

Dấu "=" xảy ra: `x-3=0<=>x=3` 

Bài 4.2:

\(a.\left(\dfrac{1}{4}\right)^3\cdot\left(\dfrac{1}{8}\right)^2\\ =\left[\left(\dfrac{1}{2}\right)^2\right]^3\cdot\left[\left(\dfrac{1}{2}\right)^3\right]^2\\ =\left(\dfrac{1}{2}\right)^6\cdot\left(\dfrac{1}{2}\right)^6\\ =\left(\dfrac{1}{2}\right)^{12}\\ b.25\cdot5^3\cdot\dfrac{1}{625}\cdot5^3\\ =5^2\cdot5^3\cdot\dfrac{1}{5^4}\cdot5^3\\ =5^8\cdot\dfrac{1}{5^4}\\ =5^4\\ c.4^2\cdot32:2^3\\ =\left(2^2\right)^2\cdot2^5:2^3\\ =2^4\cdot2^5:2^3\\ =2^{4+5-3}\\ =2^6\\ d.5^6\cdot\dfrac{1}{20}\cdot2^2\cdot3^3:125\\ =\left(\dfrac{1}{20}\cdot2^2\cdot5\right)\cdot5^5\cdot3^3:5^3\\ =5^2\cdot3^3\)

bài 4.3:

a: \(\dfrac{4^6\cdot9^5+6^9\cdot120}{8^4\cdot3^{12}-6^{11}}=\dfrac{2^{12}\cdot3^{10}+2^{12}\cdot3^{10}\cdot5}{2^{12}\cdot3^{12}-2^{11}\cdot3^{11}}\)

\(=\dfrac{2^{12}\cdot3^{10}\left(1+5\right)}{2^{11}\cdot3^{11}\left(2\cdot3-1\right)}=\dfrac{2}{3}\cdot\dfrac{6}{5}=\dfrac{12}{15}=\dfrac{4}{5}\)

b: \(\dfrac{9\cdot5^{20}\cdot27^9-3\cdot9^{15}\cdot25^9}{7\cdot3^{29}\cdot125^6-3\cdot3^9\cdot15^{19}}\)

\(=\dfrac{3^2\cdot5^{20}\cdot3^{27}-3\cdot3^{30}\cdot5^{18}}{7\cdot3^{29}\cdot5^{18}-3^{10}\cdot3^{19}\cdot5^{19}}\)

\(=\dfrac{3^{29}\cdot5^{18}\left(5^2-3^2\right)}{3^{29}\cdot5^{18}\left(7-5\right)}=\dfrac{16}{2}=8\)