Solve $\sqrt[4]{x}+\sqrt[4]{x+1}=\sqrt[4]{2x+1}$

Let $\sqrt[4]{x}=a$ and $\sqrt[4]{x+1}=b$.

Thus, $a\geq0,$ $b\geq1$, $b^4-a^4=1$ and $$a+b=\sqrt[4]{a^4+b^4}$$ or $$(a+b)^4=a^4+b^4$$ or $$2ab(2a^2+3ab+2b^2)=0,$$ which gives $$ab=0.$$ Can you end it now.

If you raise both sides to the 4th power you get

$$x + \mbox{junk} + x+1 = 2x+1$$

or

$$\mbox{junk} =0.$$

If $x>0$ then junk is positive, and you don't have a solution. So the only solution is $x=0.$

You have $x^{1/4}+(x+1)^{1/4}=(2x+1)^{1/4}$

Raise both sides to the power $4$ and you have: $$x+(x+1)+4x^{3/4}(x+1)^{1/4}+6x^{1/4}(x+1)^{1/4}+4x^{1/4}(x+1)^{3/4}=2x+1$$ $$x^{1/4}(x+1)^{1/4}\Big(2x^{1/2}+3x^{1/4}(x+1)^{1/4}+2(x+1)^{1/4}\Big)=0$$ From here we have $x=0$ or $x=-1$ as solution.

Now if $x\neq 0$ and $x\neq -1$, then $2x^{1/2}+3x^{1/4}(x+1)^{1/4}+2(x+1)^{1/4}=0$

Observe that $2x^{1/2}+3x^{1/4}(x+1)^{1/4}+2(x+1)^{1/4}=2(x^{1/2}+2x^{1/4}(x+1)^{1/4}+(x+1)^{1/2})-x^{1/4}(x+1)^{1/4}=2\Big(x^{1/4}+(x+1)^{1/4}\Big)^2-x^{1/4}(x+1)^{1/4}$

$$2\Big(x^{1/4}+(x+1)^{1/4}\Big)^2=x^{1/4}(x+1)^{1/4}$$ $$2(2x+1)^{1/2}=x^{1/4}(x+1)^{1/4}$$

Raising both the sides to the power of $4$ gives: $$16(2x+1)^2=x(x+1)$$

This in fact will give us an quadratic equation. I hope you can solve from here.