To determine the roots of the equation [tex]\( x^{\sqrt{x}} = \left(\sqrt{x}\right)^x \)[/tex], we will solve it step-by-step:
1. Understand the Equation:
The given equation is [tex]\( x^{\sqrt{x}} = \left(\sqrt{x}\right)^x \)[/tex]. Let's rewrite it for clarity.
- The left-hand side (LHS) is [tex]\( x \)[/tex] raised to the power of [tex]\( \sqrt{x} \)[/tex].
- The right-hand side (RHS) is [tex]\( \sqrt{x} \)[/tex] raised to the power of [tex]\( x \)[/tex].
2. Simplify the Right-Hand Side:
We know that [tex]\( \sqrt{x} \)[/tex] can be written as [tex]\( x^{1/2} \)[/tex]. Therefore:
[tex]\[
\left(\sqrt{x}\right)^x = \left(x^{1/2}\right)^x = x^{\frac{x}{2}}
\][/tex]
3. Equate the Exponents:
The equation now becomes:
[tex]\[
x^{\sqrt{x}} = x^{\frac{x}{2}}
\][/tex]
For the equation [tex]\( a^b = a^c \)[/tex] (assuming [tex]\( a \neq 0 \)[/tex] and [tex]\( a \neq 1 \)[/tex]), the exponents must be equal. Thus, we equate the exponents:
[tex]\[
\sqrt{x} = \frac{x}{2}
\][/tex]
4. Solve the Exponent Equation:
To solve [tex]\(\sqrt{x} = \frac{x}{2}\)[/tex], let [tex]\( y = \sqrt{x} \)[/tex]. Then, [tex]\( y^2 = x \)[/tex]. Substitute this into the exponent equation:
[tex]\[
y = \frac{y^2}{2}
\][/tex]
Multiply both sides by 2 to clear the fraction:
[tex]\[
2y = y^2
\][/tex]
Rearrange and simplify the quadratic equation:
[tex]\[
y^2 - 2y = 0
\][/tex]
Factorize:
[tex]\[
y(y - 2) = 0
\][/tex]
5. Find the Values of [tex]\( y \)[/tex]:
Set each factor to zero and solve for [tex]\( y \)[/tex]:
[tex]\[
y = 0 \quad \text{or} \quad y = 2
\][/tex]
6. Back-Substitute for [tex]\( x \)[/tex]:
Recall that [tex]\( y = \sqrt{x} \)[/tex]:
- If [tex]\( y = 0 \)[/tex], then [tex]\( \sqrt{x} = 0 \)[/tex] which implies [tex]\( x = 0 \)[/tex].
- If [tex]\( y = 2 \)[/tex], then [tex]\( \sqrt{x} = 2 \)[/tex] which implies [tex]\( x = 2^2 = 4 \)[/tex].
7. Additional Solution [tex]\( x = 1 \)[/tex]:
Testing the case [tex]\( x = 1 \)[/tex]:
- Evaluate LHS: [tex]\( 1^{\sqrt{1}} = 1^1 = 1 \)[/tex]
- Evaluate RHS: [tex]\( (\sqrt{1})^1 = 1^1 = 1 \)[/tex]
Since both sides are equal, [tex]\( x = 1 \)[/tex] is also a solution.
8. Conclusion:
Therefore, the roots of the equation [tex]\( x^{\sqrt{x}} = \left(\sqrt{x}\right)^x \)[/tex] are:
[tex]\[
\boxed{0, 1, 4}
\][/tex]
