What is the reduced radical form of each expression? Assume all variables are positive. SEE EXAMPLE 2 23. \( \sqrt[3]{250 y^{2} z^{4}} \) \( \begin{array}{ll}\text { 25. } \frac{\sqrt{48 x^{3}}}{3 x y^{2}} & \text { 24. } \sqrt[4]{256 v^{7} w^{12}} \\ \text { 27. } \sqrt[3]{216 m} & \text { 26. } \sqrt{\frac{56 x^{5} y^{5}}{7 x y}} \\ \end{array} \)
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To tackle these expressions, let's first simplify each one. For \( \sqrt[3]{250 y^{2} z^{4}} \), we can break it down: \( 250 = 2 \times 125 = 2 \times 5^3 \) and the radicals yield \( 5 \) from the \( 5^3 \), leaving \( 2 \) under the cube root, along with \( y^{2} \) and \( z^{4} = z^{3}z \), which provides \( z \) outside the root. Thus, the reduced radical form is \( 5z\sqrt[3]{2y^{2}z} \). For the expression \( \frac{\sqrt{48 x^{3}}}{3 x y^{2}} \), simplify the numerator first: \( \sqrt{48} = 4\sqrt{3} \) and \( \sqrt{x^{3}} = x\sqrt{x} \). So, we rewrite it as \( \frac{4x\sqrt{3x}}{3xy^{2}} \). Canceling out \( x \) in the numerator and denominator results in \( \frac{4\sqrt{3x}}{3y^{2}} \). Wasn't that fun? Now let's dive deeper into how these concepts are applied in everyday life! In architecture, understanding how to deal with radicals helps in calculating areas and volumes, especially when designing complex structures. Your math skills come in handy when optimizing spaces or figuring out material needs based on irregular shapes! Furthermore, awareness of common mistakes can elevate your mathematical prowess. When simplifying radicals, many forget to check for the highest perfect square or cube factor, leading to incomplete answers. Always ensure you're breaking down numbers correctly and double-check the indices for accuracy. Happy simplifying!
