\(a,-x^3+3x^2-3x+1=-\left(x^3-3x^2+3x-1\right)=-\left(x^3-3.x^2.1+3.x.1^2-1^3\right)\)

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

\(b,8-12x+6x^2-x^3=2^3-3.2^2.x+3.2.x^2-x^3=\left(2-x\right)^3\)

\(a,x^3+12x^2+48x+64=x^3+3.x^2.4+3.x.4^2+4^3=\left(x+4\right)^3=\left(6+4\right)^3=10^3=1000\)

\(b,x^3-6x^2+12x-8=x^3-3.x^2.2+3.x.2^2-2^3=\left(x-2\right)^3=\left(22-2\right)^3=20^3=8000\)

P(x)-Q(x)= 4x³-9x²+5x

1.\(45^{10}.5^{30}=45^{10}.125^{10}=\left(45.125\right)^{10}=5625^{10}\)

2.a. \(\left(2x-1\right)^3=-8\Leftrightarrow\left(2x-1\right)^3=\left(-2\right)^3\)

\(\Leftrightarrow2x-1=-2\Leftrightarrow x=-\frac{1}{2}\)

b.\(\left(x+\frac{1}{2}\right)^2=\frac{1}{16}\Leftrightarrow\orbr{\begin{cases}x+\frac{1}{2}=\frac{1}{4}\\x+\frac{1}{2}=-\frac{1}{4}\end{cases}}\)

\(\Leftrightarrow\orbr{\begin{cases}x=-\frac{1}{4}\\x=-\frac{3}{4}\end{cases}}\)

c. \(\left(2x+3\right)^2=\frac{9}{121}\Leftrightarrow\orbr{\begin{cases}2x+3=\frac{3}{11}\\2x+3=-\frac{3}{11}\end{cases}}\)

\(\Leftrightarrow\orbr{\begin{cases}x=-\frac{15}{11}\\x=-\frac{18}{11}\end{cases}}\)

d.\(\left(3x-1\right)^3=-\frac{8}{27}=\left(-\frac{2}{3}\right)^3\)

\(\Leftrightarrow3x-1=-\frac{2}{3}\Leftrightarrow x=\frac{1}{9}\)

4.

a.\(99^{20}=\left(99^2\right)^{10}=9801^{10}\)

Do \(9801^{10}< 9999^{10}\Rightarrow99^{20}< 9999^{10}\)

b.\(3^{4000}=\left(3^2\right)^{2000}=9^{2000}\)

\(\Rightarrow3^{4000}=9^{2000}\)

c.\(2^{332}=\left(2^3\right)^{110}.2^2=8^{110}.4\)

\(3^{223}=\left(3^2\right)^{110}.3^3=\left(3^2\right)^{110}.9=9^{110}.9\)

Ta thấy \(4.8^{110}< 9.9^{110}\)

Vậy \(2^{332}< 3^{223}\)

a) Ta có: \(\frac{1}{27}x^3-8y^6\)

\(=\left(\frac{1}{3}x\right)^3-\left(2y^2\right)^3\)

\(=\left(\frac{1}{3}x-2y^2\right)\left(\frac{1}{9}x^2+\frac{2}{3}xy^2+4y^4\right)\)

b) Ta có: \(t^2x^6-\frac{4}{9}y^4\)

\(=\left(tx^3\right)^2-\left(\frac{2}{3}y^2\right)^2\)

\(=\left(tx^3-\frac{2}{3}y^2\right)\left(tx^3+\frac{2}{3}y^2\right)\)

c) Ta có: \(64x^6+\frac{1}{27}y^3\)

\(=\left(4x^2\right)^3+\left(\frac{1}{3}y\right)^3\)

\(=\left(4x^2+\frac{1}{3}y\right)\left(8x^4-\frac{4}{3}x^2y+\frac{1}{9}y^2\right)\)

d) Ta có: \(\frac{1}{16}a^2x^6-y^4\)

\(=\left(\frac{1}{4}ax^3\right)^2-\left(y^2\right)^2\)

\(=\left(\frac{1}{4}ax^3-y^2\right)\left(\frac{1}{4}ax^3+y^2\right)\)

e) Ta có: \(m^4x^6-\frac{4}{25}y^2\)

\(=\left(m^2x^3\right)^2-\left(\frac{2}{5}y\right)^2\)

\(=\left(m^2x^3-\frac{2}{5}y\right)\left(m^2x^3+\frac{2}{5}y\right)\)

f) Ta có: \(27x^6-\frac{1}{64}y^3\)

\(=\left(3x^2\right)^3-\left(\frac{1}{4}y\right)^3\)

\(=\left(3x^2-\frac{1}{4}y\right)\left(9x^4+\frac{3}{4}x^2y+\frac{1}{16}y^2\right)\)