a. 7³

b. 7.35.7.25 = 7 . 5 . 7 . 7 . 5 . 5 = 7³ . 5³

c. 2.3.8.12.24 = 2.3.2.2.2.2.2.3.2.2.2.3 = 2⁹ . 3³

d. 12.12.2.12.6 = 3.2.2.3.2.2.2.3.2.2.2.3 = 3⁴ . 2⁹

e. 25.5.4.2.10 = 5.5.5.2.2.2.5.2 = 5⁴ . 2⁴

f. 2.10.10.3.5.10 = 2.2.5.2.5.3.5.2.5 = 2⁴ . 5⁴ . 3

g. a.a.a+b.b.b.b = a³ + b³

h. x.x.y.y.x.y.x = x⁴ . y³

a.\(7^3\)

b.\(7^3\cdot5^3\)

c.\(2^9\cdot3^3\)

d.\(2^8\cdot3^4\)

e.\(2^4\cdot5^4\)

f.\(2^4\cdot3\cdot5^4\)

g.\(a^3+b^4\)

h.\(x^4\cdot y^3\)

a.

b.

c.

d.

e.

f.

g.

h.

d: \(D=x^3-6x^2+12x-100\)

\(=x^3-6x^2+12x-8-92\)

\(=\left(x-2\right)^3-92\)

Khi x=-98 thì \(D=\left(-98-2\right)^3-92=-1000000-92=-1000092\)

e: \(E=\left(x+1\right)^3+6\left(x+1\right)^2+12x+20\)

\(=\left(x+1\right)^3+6\left(x+1\right)^2+12\left(x+1\right)+8\)

\(=\left(x+1+2\right)^3\)

\(=\left(x+3\right)^3\)

Khi x=5 thì \(E=\left(5+3\right)^3=8^3=512\)

f: \(F=\left(2x-1\right)\left(4x^2+2x+1\right)-7\left(x^3+1\right)\)

\(=\left(2x\right)^3-1^3-7x^3-7\)

\(=x^3-8\)

Khi x=-1/2 thì \(F=\left(-\dfrac{1}{2}\right)^3-8=-\dfrac{1}{8}-8=-\dfrac{65}{8}\)

g: \(G=\left(-x-2\right)^3+\left(2x-4\right)\left(x^2+2x+4\right)-x^2\left(x-6\right)\)

\(=-\left(x+2\right)^3+2\left(x-2\right)\left(x^2+2x+4\right)-x^3+6x^2\)

\(=-x^3-6x^2-12x-8+2\left(x^3-8\right)-x^3+6x^2\)

\(=-2x^3-12x-8+2x^3-16=-12x-24\)

Khi x=-2 thì \(G=-12\cdot\left(-2\right)-24=24-24=0\)

h: \(H=\left(x-1\right)^3-\left(x+2\right)\left(x^2-2x+4\right)+3\left(x+4\right)\left(x-4\right)\)

\(=x^3-3x^2+3x-1-\left(x^3+8\right)+3\left(x^2-16\right)\)

\(=x^3-3x^2+3x-1-x^3-8+3x^2-48\)

\(=3x-57\)

Khi x=-1/2 thì \(H=3\cdot\dfrac{-1}{2}-57=-1,5-57=-58,5\)

\(a,A=\dfrac{3x+2-3x+2+3x-6}{\left(3x-2\right)\left(3x+2\right)}=\dfrac{3x-2}{\left(3x-2\right)\left(3x+2\right)}=\dfrac{1}{3x+2}\\ b,B=\dfrac{1}{2}+\dfrac{x}{\dfrac{x+2-x}{x+2}}=\dfrac{1}{2}+\dfrac{x}{\dfrac{2}{x+2}}=\dfrac{1}{2}+\dfrac{x\left(x+2\right)}{2}\\ B=\dfrac{1+x^2+2x}{2}=\dfrac{\left(x+1\right)^2}{2}\)

\(C=\sqrt{\dfrac{x-6\sqrt{x}+9}{x+6\sqrt{x}+9}}=\sqrt{\dfrac{\left(\sqrt{x}-3\right)^2}{\left(\sqrt{x}+3\right)^2}}=\dfrac{\left|\sqrt{x}-3\right|}{\sqrt{x}+3}\)

Vì \(x\ge9\Rightarrow\sqrt{x}\ge3\Leftrightarrow\sqrt{x}-3\ge0\)

\(\Rightarrow C=\dfrac{\sqrt{x}-3}{\sqrt{x}+3}\)

\(D=\dfrac{x-1}{\sqrt{y}-1}.\sqrt{\dfrac{y-2\sqrt{y}+1}{\left(x-1\right)^4}}\) (\(x;y\ne1;y\ge0\))

\(=\dfrac{x-1}{\sqrt{y}-1}.\dfrac{\sqrt{\left(\sqrt{y}-1\right)^2}}{\left(x-1\right)^2}=\dfrac{\left|\sqrt{y}-1\right|}{\left(\sqrt{y}-1\right)\left(x-1\right)}\)

TH1: \(\sqrt{y}-1>0\Leftrightarrow y>1\)

\(\Rightarrow D=\dfrac{\sqrt{y}-1}{\left(\sqrt{y}-1\right)\left(x-1\right)}=\dfrac{1}{x-1}\)

TH2:\(\sqrt{y}-1< 0\Leftrightarrow0\le y< 1\)

\(\Rightarrow D=\dfrac{-\left(\sqrt{y}-1\right)}{\left(\sqrt{y}-1\right)\left(x-1\right)}=\dfrac{-1}{x-1}\)

Vậy...

\(E=\dfrac{1}{2x-1}.\sqrt{5x^4\left(1-4x+4x^2\right)}\) 

\(=\dfrac{1}{2x-1}\sqrt{5x^4\left(2x-1\right)^2}=\dfrac{\sqrt{5}x^2\left|2x-1\right|}{2x-1}\)

TH1: \(2x-1>0\Leftrightarrow x>\dfrac{1}{2}\)

\(\Rightarrow E=\dfrac{\sqrt{5}x^2\left(2x-1\right)}{2x-1}=\sqrt{5}x^2\)

TH2:\(2x-1< 0\Leftrightarrow x< \dfrac{1}{2}\)

\(\Rightarrow E=\dfrac{-\sqrt{5}x^2\left(2x-1\right)}{2x-1}=-\sqrt{5}x^2\)

Vậy...

c)

\(C=\sqrt{\dfrac{x-6\sqrt{x}+9}{x+6\sqrt{x}+9}}=\sqrt{\dfrac{\left(\sqrt{x}-3\right)^2}{\left(\sqrt{x}+3\right)^2}}=\dfrac{\sqrt{x}-3}{\sqrt{x}+3}\)

d)

\(D=\dfrac{x-1}{\sqrt{y}-1}.\sqrt{\dfrac{y-2\sqrt{y}+1}{\left(x-1\right)^4}}=\dfrac{x-1}{\sqrt{y}-1}.\dfrac{\left|\sqrt{y}-1\right|}{\left(x-1\right)^2}=\dfrac{\left|\sqrt{y}-1\right|}{\left(\sqrt{y}-1\right)\left(x-1\right)}\)

e)

\(E=\dfrac{1}{2x-1}.\sqrt{5x^4\left(1-4x+4x^2\right)}=\dfrac{1}{2x-1}.\sqrt{5x^4.\left(2x-1\right)^2}=\dfrac{1}{2x-1}.\sqrt{5}x^2.\left|2x-1\right|\)

\(\sqrt{\dfrac{x}{y}}+\sqrt{xy}+\dfrac{x}{y}\sqrt{\dfrac{y}{x}}=\sqrt{\dfrac{x}{y}}+\sqrt{\dfrac{x}{y}}\cdot\sqrt{y^2}+\sqrt{\dfrac{x}{y}}\cdot\sqrt{\dfrac{x}{y}\cdot\dfrac{y}{x}}=\sqrt{\dfrac{x}{y}}\cdot\left(1+y+1\right)=\sqrt{\dfrac{x}{y}}\cdot\left(y+2\right)\)

Ta có: \(\sqrt{\dfrac{x}{y}}+\sqrt{xy}+\dfrac{x}{y}\cdot\sqrt{\dfrac{y}{x}}\)

\(=\dfrac{\sqrt{x}}{\sqrt{y}}+\dfrac{\sqrt{x}}{\sqrt{y}}+\sqrt{xy}\)

\(=\dfrac{2\sqrt{x}}{\sqrt{y}}+\dfrac{y\sqrt{x}}{\sqrt{y}}\)

\(=\dfrac{2\sqrt{x}+y\sqrt{x}}{\sqrt{y}}\)

\(1,\dfrac{\sqrt{27\left(x-5\right)^2}}{\sqrt{3}}\left(dkxd:x\ge5\right)\)

\(=\dfrac{\sqrt{27}.\sqrt{\left(x-5\right)^2}}{\sqrt{3}}\)

\(=\dfrac{\sqrt{3}.\sqrt{3^2}.\left|x-5\right|}{\sqrt{3}}\)

\(=3\left(x-5\right)\)

\(=3x-15\)

\(2,\dfrac{\sqrt{\left(x-4\right)^2}}{\sqrt{9\left(x-4\right)^2}}\left(dkxd:x< 4\right)\)

\(=\dfrac{\left|x-4\right|}{\sqrt{9}.\left|x-4\right|}\)

\(=\dfrac{1}{\sqrt{3}^2}\)

\(=\dfrac{1}{3}\)

\(1,\dfrac{\sqrt{27\left(x-5\right)^2}}{\sqrt{3}}\\ =\dfrac{\sqrt{27}.\sqrt{\left(x-5\right)^2}}{\sqrt{3}}\\ =\dfrac{3\sqrt{3}.\left|x-5\right|}{\sqrt{3}}=3.\left(x-5\right)=3-15\\ 2,\dfrac{\sqrt{\left(x-4\right)^2}}{\sqrt{9\left(x-4\right)^2}}\\ =\dfrac{\left|x-4\right|}{\sqrt{9}.\sqrt{\left(x-4\right)^2}}\\ =\dfrac{\left|x-4\right|}{\sqrt{9}.\left|x-4\right|}=\dfrac{4-x}{3.\left(4-x\right)}=\dfrac{1}{3}\)

`@` `\text {Ans}`

`\downarrow`

`a)`

\((x+3)^2 - (x-4)(x+4)\)

`= (x^2 + 6x + 9) - (x^2 + 4x - 4x - 16)`

`= x^2 + 6x + 9 - (x^2 - 16)`

`= x^2 + 6x + 9 - x^2 + 16`

`= 6x + 25`

`b)`

\(3(x+2)^2 + (2x-1)^2 - 7(3+x)(x-3)\)

`= 3(x^2 + 4x + 4) + (4x^2 - 4x + 1) - 7(x^2 - 9)`

`= 3x^2 + 12x + 12 + 4x^2 - 4x + 1 - 7x^2 + 63`

`= (3x^2 + 4x^2 - 7x^2) + (12x - 4x) + (12 + 1 + 63)`

`= 8x + 76`

`c)`

\((x+2)^2 + (x-3)^2 - 2(x+1)(x-1)\)

`= (x^2 + 4x + 4) + (x^2 - 6x + 9) - 2(x^2 - 1)`

`= x^2 + 4x + 4 + x^2 - 6x + 9 - 2x^2 + 2`

`= (x^2 + x^2 - 2x^2) + (4x - 6x) + (4 + 9 + 2)`

`= -2x + 15`

`d)`

\((x+1)^3 - x(x-2)^2 + x - 1\)

`= (x^3 + 3x^2 + 3x + 1) - x(x^2 - 4x + 4) + x - 1`

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

`= (x^3 - x^3) + (3x^2 + 4x^2) + (3x - 4x + x) + (1-1)`

`= 7x^2 `

`@` `\text {Kaizuu lv uuu}`

\(B=cos\left(\pi+\dfrac{\pi}{2}-a\right)-sin\left(8\pi-a\right)+tan\left(3\pi+\dfrac{\pi}{2}-a\right)+cot\left(3\pi-a\right)\)

\(=-cos\left(\dfrac{\pi}{2}-a\right)-sin\left(-a\right)+tan\left(\dfrac{\pi}{2}-a\right)+cot\left(-a\right)\)

\(=-sina+sina+cota-cota=0\)

\(C=sin\left(a+\dfrac{\pi}{2}+42\pi\right)-cos\left(2022\pi-a\right)+sin^2\left(33\pi+a\right)+sin^2\left(a-\dfrac{\pi}{2}-2\pi\right)\)

\(=sin\left(a+\dfrac{\pi}{2}\right)-cosa+sin^2a+sin^2\left(\dfrac{\pi}{2}-a\right)\)

\(=cosa-cosa+sin^2a+cos^2a=1\)