Find the values of x when,
x raised to the power root x = x raised to the power x whole under root.
To find the values of x, we need to solve the equation:
x^(√x) = x^(x/√x)
To begin, let's simplify the equation by using the property of exponents that states: a^(b/c) = (a^b)^(1/c)
Our equation now becomes:
(x^√x)^√x = (x^(x/√x))^(1/√x)
We can simplify each side separately:
(x^√x)^(√x) = (x^(x/√x))^(1/√x)
Now, since the bases of both sides are equal, we can equate their exponents:
√x * √x = x/√x
Simplifying this equation further:
x = x/√x
To eliminate the square root in the denominator, we can multiply both sides of the equation by √x:
√x * x = x
Simplifying:
x^(3/2) = x
Now we have an equation with a common base on both sides. We can equate the exponents:
3/2 = 1
However, this is not possible since 3/2 is not equal to 1. Therefore, there are no solutions for this equation.
In summary, there are no values of x that satisfy the equation x^(√x) = x^(x/√x).