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$x^x=0.5^{0.5}$

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Solve exponential equations by rewriting the base

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Q.

x^x=0.5^{0.5}

Recognize equation form: Recognize the form of the equation. $\newline$ The equation is of the form $x^x = (0.5)^{0.5}$ . We need to find the value of $x$ that satisfies this equation.
Simplify right side: Simplify the right side of the equation. $\newline$ We know that $0.5$ is the same as $\frac{1}{2}$ , and the square root of $\frac{1}{2}$ is the same as $(\frac{1}{2})^{\frac{1}{2}}$ . So, $(0.5)^{0.5}$ is the same as $(\frac{1}{2})^{\frac{1}{2}}$ .
Set up simplified equation: Set up the equation with the simplified right side. $\newline$ Now we have $x^x = (\frac{1}{2})^{\frac{1}{2}}$ .
Find solution pattern: Look for a pattern or a property that can help solve the equation. $\newline$ Notice that if $x$ were equal to $\frac{1}{2}$ , then the left side of the equation would be $\left(\frac{1}{2}\right)^{\frac{1}{2}}$ , which is the same as the right side. So, it seems that $x = \frac{1}{2}$ might be the solution.
Verify solution: Verify the solution. $\newline$ If $x = \frac{1}{2}$ , then $x^x = (\frac{1}{2})^{\frac{1}{2}}$ . This is true because any number raised to the power of itself where the number is the square root of that power will equal the number. Therefore, $(\frac{\(1\)}{\(2\)})^{\frac{\(1\)}{\(2\)}} = (\frac{\(1\)}{\(2\)})^{\frac{\(1\)}{\(2\)}}.

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