10. $$ x^{2}-3 x^{\frac{1}{4}}+2=0 $$ 13. $$ x-x^{\frac{1}{2}}-20=0 $$ 16. $$ x-3 x^{\frac{1}{2}}-18=0 $$ 19. $$ x^{\frac{2}{3}}-3 x^{\frac{1}{3}}-4=0 $$

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Solve the equation $$ x^{2}-3x^{\frac{1}{4}}+2=0 $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$x^{2}-3x^{\frac{1}{4}}+2=0$$
- step1: Find the domain:
  $$x^{2}-3x^{\frac{1}{4}}+2=0,x\geq 0$$
- step2: Factor the expression:
  $$\left(x^{\frac{1}{4}}-1ight)\left(x^{\frac{3}{4}}+x^{\frac{1}{4}}-1ight)\left(x+x^{\frac{3}{4}}+x^{\frac{1}{4}}+2ight)=0$$
- step3: Separate into possible cases:
  $$\begin{align}&x^{\frac{1}{4}}-1=0\&x^{\frac{3}{4}}+x^{\frac{1}{4}}-1=0\&x+x^{\frac{3}{4}}+x^{\frac{1}{4}}+2=0\end{align}$$
- step4: Solve the equation:
  $$\begin{align}&x=1\&x\approx 0.216757\&x 
otin \mathbb{R}\end{align}$$
- step5: Find the union:
  $$\begin{align}&x=1\&x\approx 0.216757\end{align}$$
- step6: Check if the solution is in the defined range:
  $$\begin{align}&x=1\&x\approx 0.216757\end{align},x\geq 0$$
- step7: Find the intersection:
  $$\begin{align}&x=1\&x\approx 0.216757\end{align}$$
- step8: Rewrite:
  $$x_{1}\approx 0.216757,x_{2}=1$$
Solve the equation $$ x-x^{\frac{1}{2}}-20=0 $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$x-x^{\frac{1}{2}}-20=0$$
- step1: Find the domain:
  $$x-x^{\frac{1}{2}}-20=0,x\geq 0$$
- step2: Solve using substitution:
  $$t^{2}-t-20=0$$
- step3: Factor the expression:
  $$\left(t-5ight)\left(t+4ight)=0$$
- step4: Separate into possible cases:
  $$\begin{align}&t-5=0\&t+4=0\end{align}$$
- step5: Solve the equation:
  $$\begin{align}&t=5\&t=-4\end{align}$$
- step6: Substitute back:
  $$\begin{align}&x^{\frac{1}{2}}=5\&x^{\frac{1}{2}}=-4\end{align}$$
- step7: Solve the equation for $$x:$$
  $$\begin{align}&x=25\&x 
otin \mathbb{R}\end{align}$$
- step8: Find the union:
  $$x=25$$
- step9: Check if the solution is in the defined range:
  $$x=25,x\geq 0$$
- step10: Find the intersection:
  $$x=25$$
Solve the equation $$ x-3x^{\frac{1}{2}}-18=0 $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$x-3x^{\frac{1}{2}}-18=0$$
- step1: Find the domain:
  $$x-3x^{\frac{1}{2}}-18=0,x\geq 0$$
- step2: Solve using substitution:
  $$t^{2}-3t-18=0$$
- step3: Factor the expression:
  $$\left(t-6ight)\left(t+3ight)=0$$
- step4: Separate into possible cases:
  $$\begin{align}&t-6=0\&t+3=0\end{align}$$
- step5: Solve the equation:
  $$\begin{align}&t=6\&t=-3\end{align}$$
- step6: Substitute back:
  $$\begin{align}&x^{\frac{1}{2}}=6\&x^{\frac{1}{2}}=-3\end{align}$$
- step7: Solve the equation for $$x:$$
  $$\begin{align}&x=36\&x 
otin \mathbb{R}\end{align}$$
- step8: Find the union:
  $$x=36$$
- step9: Check if the solution is in the defined range:
  $$x=36,x\geq 0$$
- step10: Find the intersection:
  $$x=36$$
Solve the equation $$ x^{\frac{2}{3}}-3x^{\frac{1}{3}}-4=0 $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$x^{\frac{2}{3}}-3x^{\frac{1}{3}}-4=0$$
- step1: Solve using substitution:
  $$t^{2}-3t-4=0$$
- step2: Factor the expression:
  $$\left(t-4ight)\left(t+1ight)=0$$
- step3: Separate into possible cases:
  $$\begin{align}&t-4=0\&t+1=0\end{align}$$
- step4: Solve the equation:
  $$\begin{align}&t=4\&t=-1\end{align}$$
- step5: Substitute back:
  $$\begin{align}&x^{\frac{1}{3}}=4\&x^{\frac{1}{3}}=-1\end{align}$$
- step6: Solve the equation for $$x:$$
  $$\begin{align}&x=64\&x=-1\end{align}$$
- step7: Rewrite:
  $$x_{1}=-1,x_{2}=64$$
The solutions to the given equations are:
1. $$ x^{2}-3x^{\frac{1}{4}}+2=0 $$ has solutions $$ x_{1} \approx 0.216757 $$ and $$ x_{2} = 1 $$.
2. $$ x-x^{\frac{1}{2}}-20=0 $$ has solution $$ x = 25 $$.
3. $$ x-3x^{\frac{1}{2}}-18=0 $$ has solution $$ x = 36 $$.
4. $$ x^{\frac{2}{3}}-3x^{\frac{1}{3}}-4=0 $$ has solutions $$ x_{1} = -1 $$ and $$ x_{2} = 64 $$.
The solutions to the equations are: 1. $$ x \approx 0.216757 $$ and $$ x = 1 $$ 2. $$ x = 25 $$ 3. $$ x = 36 $$ 4. $$ x = -1 $$ and $$ x = 64 $$

10. \( x^{2}-3 x^{\frac{1}{4}}+2=0 \) 13. \( x-x^{\frac{1}{2}}-20=0 \) 16. \( x-3 x^{\frac{1}{2}}-18=0 \) 19. \( x^{\frac{2}{3}}-3 x^{\frac{1}{3}}-4=0 \)

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