sqrt(-(-1)^(|__(1)/(2)+(1)/(pi)__|)*(-pi*|__(1)/(2)+(1)/(pi)__|+1))

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Simplify Absolute Value: We need to simplify the expression inside the square root first. Let's start by simplifying the absolute value expression |1/2 + 1/pi|. Calculate the value inside the absolute value. 1/2 + 1/pi = (pi/2 + 1)/pi Since pi is approximately 3.14159, which is greater than 2, the numerator (pi/2 + 1) is positive. Therefore, the absolute value of a positive number is the number itself. |1/2 + 1/pi| = |(pi/2 + 1)/pi| = (pi/2 + 1)/piSimplify Exponent: Now, let's simplify the exponent (-1)^(|1/2 + 1/pi|). Substitute the absolute value we found into the exponent. (-1)^(|1/2 + 1/pi|) = (-1)^((pi/2 + 1)/pi) Since the exponent is a rational number and not an integer, the result will be complex. However, the base is -1, and any power of -1 that is not an integer will result in a complex number with a magnitude of 1.Simplify Term: Next, we need to simplify the term (-pi*|1/2 + 1/pi|+1). Substitute the absolute value we found into the expression. (-pi*|1/2 + 1/pi|+1) = (-pi*((pi/2 + 1)/pi)+1) Simplify the expression. = (-pi*(pi/2 + 1)/pi + 1) = (-(pi/2 + 1) + 1) = (-pi/2 - 1 + 1) = -pi/2Rewrite Original Expression: Now we have the simplified terms, we can rewrite the original expression with these simplified parts. sqrt(-(-1)^((pi/2 + 1)/pi)*(-pi/2)) Since (-1)^((pi/2 + 1)/pi) is a complex number with a magnitude of 1, we can denote it as cis(theta) where cis(theta) represents cos(theta) + i*sin(theta) and theta is some angle. The negative sign in front of it will change the direction of the angle, but the magnitude will still be 1.Multiply Complex Number: We can now multiply the complex number cis(theta) by -pi/2. The multiplication of a complex number with a real number will only affect the magnitude, not the direction (angle). Since the magnitude of cis(theta) is 1, the magnitude of the product will be pi/2.Take Square Root: Finally, we take the square root of the product. sqrt(-(-1)^((pi/2 + 1)/pi)*(-pi/2)) = sqrt(-cis(theta)*(-pi/2)) Since the magnitude of cis(theta) is 1, and we are multiplying by -pi/2, the magnitude under the square root is pi/2. The square root of a negative real number is a complex number. Therefore, the result will be a complex number with a magnitude of sqrt(pi/2).Express in Terms of i: The final step is to express the complex number in terms of i, the imaginary unit. sqrt(-pi/2) = i*sqrt(pi/2) This is because the square root of a negative number is imaginary.

$\sqrt{-(-1)^{\left\lfloor\frac{1}{2}+\frac{1}{\pi}\right\rfloor} \cdot\left(-\pi \cdot\left\lfloor\frac{1}{2}+\frac{1}{\pi}\right\rfloor+1\right)}$

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$\sqrt{-(-1)^{\left\lfloor\frac{1}{2}+\frac{1}{\pi}\right\rfloor} \cdot\left(-\pi \cdot\left\lfloor\frac{1}{2}+\frac{1}{\pi}\right\rfloor+1\right)}$