Answer:
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Step-by-step explanation:
To solve the equation \( x^{x^{\sqrt{x}}} = (x\sqrt{x})^x \), we will proceed step by step.
First, simplify the right-hand side:
\[ (x\sqrt{x})^x = x^x \cdot (\sqrt{x})^x = x^x \cdot x^{x/2} = x^{x + x/2} = x^{3x/2}. \]
Now, equate the simplified expressions:
\[ x^{x^{\sqrt{x}}} = x^{3x/2}. \]
Since the bases are the same (both sides are powers of \( x \)), equate the exponents:
\[ x^{\sqrt{x}} = \frac{3x}{2}. \]
To solve this equation, take the logarithm on both sides:
\[ \sqrt{x} \log x = \log \left( \frac{3x}{2} \right). \]
Let's consider possible values of \( x \) to find a solution:
**Testing \( x = 1 \):**
\[ \sqrt{1} \log 1 = \log \left( \frac{3 \cdot 1}{2} \right) \]
\[ 0 \cdot 0 = \log \left( \frac{3}{2} \right) \]
\[ 0 = 0 \quad (\text{True}). \]
Therefore, \( x = 1 \) satisfies the original equation \( x^{x^{\sqrt{x}}} = (x\sqrt{x})^x \).
**Verification for other values:**
For \( x > 1 \):
- \( x^{\sqrt{x}} \) grows slower than \( x^{3x/2} \), hence no solution.
For \( x < 1 \):
- \( x^{\sqrt{x}} \) decreases faster than \( x^{3x/2} \), hence no solution.
Therefore, the only solution to the equation \( x^{x^{\sqrt{x}}} = (x\sqrt{x})^x \) is \( \boxed{1} \).
Answer:
given
x^x^√x = (x√x)^x then x is equal to
so , x*x√x
