SOLUTION: the square root of x minus 3 square root of x with an index of 4 minus 4 equals 0

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Question 201492This question is from textbook precalculus
: the square root of x minus 3 square root of x with an index of 4 minus 4 equals 0 This question is from textbook precalculus

Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
The easiest way to do this is based on understanding that sqrt%28x%29+=+%28root%284%2C+x%29%29%5E2. This is not hard to see if you understand fractional exponents: sqrt%28x%29+=+x%5E%281%2F2%29 and root%284%2C+x%29+=+x%5E%281%2F4%29. So %28root%284%2C+x%29%29%5E2+=+%28x%5E%281%2F4%29%29%5E2. Using the appropriate rule for exposnents (%28a%5Eb%29%5Ec+=+a%5E%28b%2Ac%29

If we let y+=+root%284%2C+x%29 then y%5E2+=+sqrt%28x%29. Substituting these into
sqrt%28x%29+-3%2A%28root%284%2Cx%29%29+-4+=+0 we get
y%5E2+-+3y+-+4+=+0
This equation can be easily solved by factoring:
%28y+-+4%29%2A%28y+%2B+1%29+=+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%284%2C+x%29+=+4 or root%284%2C+x%29+=+-1
We reject the second equation because by definition root%284%2C+x%29 must not be negative. To solve the first equation we raise both sides to the 4th power:
%28root%284%2C+x%29%29%5E4+=+4%5E4 which gives:
x = 256
Checking:
sqrt%28256%29+-+3%2Aroot%284%2C+256%29+-+4+=+0
16 - 3*4 - 4 = 0
16 - 12 - 4 = 0
0 = 0 Check!!