Question 201492
The easiest way to do this is based on understanding that {{{sqrt(x) = (root(4, x))^2}}}. This is not hard to see if you understand fractional exponents: {{{sqrt(x) = x^(1/2)}}} and {{{root(4, x) = x^(1/4)}}}. So {{{(root(4, x))^2 = (x^(1/4))^2}}}. Using the appropriate rule for exposnents ({{{(a^b)^c = a^(b*c)}}} {{{(x^(1/4))^2 = x^((1/4)*2) = x^(2/4) = x^(1/2) = sqrt(x)}}}<br>
If we let {{{y = root(4, x)}}} then {{{y^2 = sqrt(x)}}}. Substituting these into
{{{sqrt(x) -3*(root(4,x)) -4 = 0}}} we get
{{{y^2 - 3y - 4 = 0}}}
This equation can be easily solved by factoring:
{{{(y - 4)*(y + 1) = 0}}}
Since a product of zero means one of the factors must be zero:
y-4=0  or  y+1=0
Adding 4 to both sides of the first equation and subtracting 1 from both sides of the second we get:
y=4 or y=-1
Now we substitute back in for y:
{{{root(4, x) = 4}}} or {{{root(4, x) = -1}}}
We reject the second equation because by definition {{{root(4, x)}}} must not be negative. To solve the first equation we raise both sides to the 4th power:
{{{(root(4, x))^4 = 4^4}}} which gives:
x = 256

Checking:
{{{sqrt(256) - 3*root(4, 256) - 4 = 0}}}
16 - 3*4 - 4 = 0
16 - 12 - 4 = 0
0 = 0 Check!!