Question

1. \(\sqrt[3]{x^2 + 2x + 1} = \sqrt[3]{x^2} + x\).

2. \(\sqrt[3]{x^3 + 2x^2 + 1} = \sqrt[3]{x + 3}\).

3. \(\sqrt[3]{x^4 - 3x^2 + 1} = \sqrt[3]{1 - 2x^3}\).

4. \(\sqrt[3]{x(x^3 + 1)} = \sqrt[3]{x^3(x + 1)}\).

5. \(\sqrt[3]{(x + 1)^2(x^2 - x + 1)} = \sqrt[3]{(x^2 + x)(x^2 + 3)}\).

Answer 1

1: \(\sqrt[3]{x^2+2x+1}=\sqrt[3]{x^2+x}\)

=>\(x^2+2x+1=x^2+x\)

=>2x+1=x

=>2x-x=-1

=>x=-1

2: \(\sqrt[3]{x^3+2x^2+1}=\sqrt[3]{x+3}\)

=>\(x^3+2x^2+1=x+3\)

=>\(x^3+2x^2-x-2=0\)

=>\(x^2\left(x+2\right)-\left(x+2\right)=0\)

=>\(\left(x+2\right)\cdot\left(x^2-1\right)=0\)

=>(x+2)(x-1)(x+1)=0

=>\(\left[\begin{array}{l}x+2=0\\ x-1=0\\ x+1=0\end{array}\right.\Rightarrow\left[\begin{array}{l}x=-2\\ x=1\\ x=-1\end{array}\right.\)

3: \(\sqrt[3]{x^4-3x^2+1}=\sqrt[3]{1-2x^3}\)

=>\(x^4-3x^2+1=1-2x^3\)

=>\(x^4+2x^3-3x^2=0\)

=>\(x^2\left(x^2+2x-3\right)=0\)

=>\(x^2\left(x+3\right)\left(x-1\right)=0\)

=>\(\left[\begin{array}{l}x^2=0\\ x+3=0\\ x-1=0\end{array}\right.\Rightarrow\left[\begin{array}{l}x=0\\ x=-3\\ x=1\end{array}\right.\)

4: \(\sqrt[3]{x\left(x^3+1\right)}=\sqrt[3]{x^3\left(x+1\right)}\)

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

=>\(x^4+x=x^4+x^3\)

=>\(x^3-x=0\)

=>x(x-1)(x+1)=0

=>\(\left[\begin{array}{l}x=0\\ x-1=0\\ x+1=0\end{array}\right.\Rightarrow\left[\begin{array}{l}x=0\\ x=1\\ x=-1\end{array}\right.\)

5: \(\sqrt[3]{\left(x+1\right)^2\cdot\left(x^2-x+1\right)}=\sqrt[3]{\left(x^2+x\right)\left(x^2+3\right)}\)

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

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

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

=>\(\left(x+1\right)\left(x^3+3x^2-x^3-1\right)=0\)

=>\(\left(x+1\right)\left(3x^2-1\right)=0\)

=>\(\left[\begin{array}{l}x+1=0\\ 3x^2-1=0\end{array}\right.\Rightarrow\left[\begin{array}{l}x=-1\\ 3x^2=1\end{array}\right.\Rightarrow\left[\begin{array}{l}x=-1\\ x^2=\frac13\end{array}\right.\Rightarrow\left[\begin{array}{l}x=-1\\ x=\frac{\sqrt3}{3}\\ x=-\frac{\sqrt3}{3}\end{array}\right.\)