3. Reduce la siguiente expresión \( M=\sqrt[7]{7^{7-7^{3}}} \cdot \sqrt[7]{\sqrt[3]{7^{4}}} \) \( \begin{array}{lll}\text { a. } 7 & \text { c. } 9 & \text { e. } 11 \\ \text { b. } 8 & \text { d. } 10 & \end{array} \)

Okay, I need to simplify the expression \( M = \sqrt[7]{7^{7 - 7^{3}}} \cdot \sqrt[7]{\sqrt[3]{7^{4}}} \) and determine which of the given options (a. 7, b. 8, c. 9, d. 10, e. 11) is the correct answer. Let's break this down step by step. First, let's simplify the exponents inside the radicals. The expression inside the first radical is \( 7^{7 - 7^{3}} \). Calculating \( 7^{3} \) gives 343, so \( 7 - 343 = -336 \). Therefore, the first radical becomes \( \sqrt[7]{7^{-336}} \). Next, the second radical is \( \sqrt[7]{\sqrt[3]{7^{4}}} \). The inner radical \( \sqrt[3]{7^{4}} \) simplifies to \( 7^{\frac{4}{3}} \). So, the second radical becomes \( \sqrt[7]{7^{\frac{4}{3}}} \). Now, the expression for \( M \) is: \[ M = \sqrt[7]{7^{-336}} \cdot \sqrt[7]{7^{\frac{4}{3}}} \] To simplify this, I can combine the exponents since both radicals have the same root (7th root). When multiplying like radicals, the exponents add: \[ M = 7^{-336 \times \frac{1}{7}} \cdot 7^{\frac{4}{3} \times \frac{1}{7}} \] \[ M = 7^{-48} \cdot 7^{\frac{4}{21}} \] Now, adding the exponents: \[ M = 7^{-48 + \frac{4}{21}} \] \[ M = 7^{-\frac{1008}{21} + \frac{4}{21}} \] \[ M = 7^{-\frac{1004}{21}} \] This simplifies to: \[ M = 7^{-\frac{1004}{21}} \] Calculating \( -\frac{1004}{21} \) gives approximately -47.81. Therefore: \[ M = 7^{-47.81} \] Since \( 7^{-47.81} \) is a very small number, it's not among the provided options. This suggests there might be a miscalculation or a different approach is needed. Let me revisit the original expression: \[ M = \sqrt[7]{7^{7 - 7^{3}}} \cdot \sqrt[7]{\sqrt[3]{7^{4}}} \] Perhaps I should consider the properties of exponents differently. Let's express both radicals with the same exponent base: \[ \sqrt[7]{7^{7 - 7^{3}}} = 7^{\frac{7 - 7^{3}}{7}} = 7^{1 - 7^{2}} \] \[ \sqrt[7]{\sqrt[3]{7^{4}}} = 7^{\frac{4}{3} \times \frac{1}{7}} = 7^{\frac{4}{21}} \] Now, multiplying these: \[ M = 7^{1 - 7^{2}} \cdot 7^{\frac{4}{21}} \] \[ M = 7^{1 - 49} \cdot 7^{\frac{4}{21}} \] \[ M = 7^{-48} \cdot 7^{\frac{4}{21}} \] \[ M = 7^{-48 + \frac{4}{21}} \] \[ M = 7^{-\frac{1008}{21} + \frac{4}{21}} \] \[ M = 7^{-\frac{1004}{21}} \] This confirms the earlier result. Given that \( 7^{-\frac{1004}{21}} \) is an extremely small number, it's clear that none of the provided options (7, 8, 9, 10, 11) are correct. There might be an error in the problem statement or the options provided. Alternatively, if there's a different interpretation or a simplification step I'm missing, I should consider that. However, based on the current calculations, none of the options a. 7, b. 8, c. 9, d. 10, e. 11 are valid answers for the given expression.