Find the values of x when,
x raised to the power root x = x
raised to the power x whole under
root.
we want
x^(√x) = √(x^x)
x^(√x) = x^(x/2)
equating the powers,
√x = x/2
x = 4
check:
x^2 = √x^4 - yes
To find the values of x in the given equation, we need to solve the equation:
x^(√x) = (x^x)^(1/√x)
Let's simplify the equation step by step:
Step 1: Simplify the right side of the equation.
(x^x)^(1/√x) = x^(x/√x) = x^(√x)
Now, we have the equation:
x^(√x) = x^(√x)
Step 2: Equate the exponents.
Since the bases are the same and both are equal to x^(√x), we can equate the exponents:
√x = x
Step 3: Square both sides of the equation.
(√x)^2 = x^2
Simplifying further:
x = x^2
Step 4: Rearrange the equation without the exponent.
x^2 - x = 0
Step 5: Factorize the equation.
x(x - 1) = 0
Now, we have two cases:
Case 1: x = 0
Case 2: x - 1 = 0 => x = 1
So, the values of x that satisfy the equation are x = 0 and x = 1.