Answer
$x=256$
Work Step by Step
$\sqrt{x}-3\sqrt[4]{x}-4=0$
Rewrite this equation using rational exponents:
$x^{1/2}-3x^{1/4}-4=0$
Let $u$ be equal to $x^{1/4}$:
$u=x^{1/4}$ so $u^{2}=x^{1/2}$
Substitute $x^{1/4}$ by $u$ and $x^{1/2}$ by $u^{2}$ in the original equation:
$u^{2}-3u-4=0$
Solve this equation by factoring:
$(u+1)(u-4)=0$
Set both factors equal to $0$ and solve each individual equation for $u$:
$u+1=0$
$u=-1$
$u-4=0$
$u=4$
Substitute $u$ back to $x^{1/4}$ and solve for $x$:
$u=-1$
$x^{1/4}=-1$
$x=(-1)^{4}$
$x=1$
$u=4$
$x^{1/4}=4$
$x=4^{4}$
$x=256$
The solutions found are $x=1$ and $x=256$. Check them by plugging them into the original equation:
$x=1$
$\sqrt{1}-3\sqrt[4]{1}-4=0$
$1-3-4=0$
$1-7=0$
$-6\ne0$ False
$x=256$
$\sqrt{256}-3\sqrt[4]{256}-4=0$
$16-3(4)-4=0$
$16-12-4=0$
$0=0$ True
The final answer is $x=256$
