Write the expression in simplest form. Assume that all variables are positive. (-32x^-10)^1/5 (-32x^-10)^1/5= square

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Write the expression in simplest form. Assume that all variables are positive. (-32x^-10)^1/5 (-32x^-10)^1/5= square

Answer 1

To simplify the expression \(\left(-32 x^{-10}\right)^{i / 5}\), we will apply the exponent rule \((a^m)^n = a^{mn}\) to both the numerical and variable parts of the expression separately.Given that \(i\) is the imaginary unit, we have:\[\left(-32\right)^{i / 5} \cdot \left(x^{-10}\right)^{i / 5}\]Now, we simplify each part:1. For the numerical part, \((-32)^{i / 5}\), we can't simplify this further without knowing the context in which the imaginary unit is used, as it will involve complex numbers.2. For the variable part, \((x^{-10})^{i / 5}\), we apply the exponent rule:\[(x^{-10})^{i / 5} = x^{-10 \cdot i / 5} = x^{-2i}\]Therefore, the expression in simplest form is:\[(-32)^{i / 5} \cdot x^{-2i}\]This is the expression simplified, but it should be noted that it involves complex numbers due to the presence of the imaginary unit \(i\).

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Write the expression in simplest form. Assume that all variables are positive. (-32x^-10)^1/5 (-32x^-10)^1/5= square

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