Simplify the following expression. Assume that the variables represent positive real numbers. \( \frac{\left(x^{1 / 5}\right)^{2}}{\left(x^{2}\right)^{7 / 5}} \) \( \frac{\left(x^{1 / 5}\right)^{2}}{\left(x^{2}\right)^{7 / 5}}=\square \) (Type exponential notation with positive exponents. Simplify your answer. Use integers or fractions for any numbers in the expression.)
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Let's simplify the expression step by step. Starting with the given expression: \[ \frac{\left(x^{1 / 5}\right)^{2}}{\left(x^{2}\right)^{7 / 5}} \] We simplify the numerator first: \[ \left(x^{1 / 5}\right)^{2} = x^{(1/5) \cdot 2} = x^{2/5} \] Next, we simplify the denominator: \[ \left(x^{2}\right)^{7 / 5} = x^{2 \cdot (7/5)} = x^{14/5} \] Now, we can rewrite the expression: \[ \frac{x^{2/5}}{x^{14/5}} \] Using the property of exponents \( \frac{a^m}{a^n} = a^{m-n} \), we get: \[ x^{(2/5) - (14/5)} = x^{(2 - 14)/5} = x^{-12/5} \] Since we need positive exponents, we can express \(x^{-12/5}\) as: \[ \frac{1}{x^{12/5}} \] Thus, the simplified expression is: \[ \frac{1}{x^{12/5}} \] Final Output: \(\frac{1}{x^{12 / 5}}\)
