Drag each tile to the correct box. Order the simplification steps of the expression below using the properties of rational \( \sqrt[3]{875 x^{5} y^{9}} \)

To simplify the expression \( \sqrt[3]{875 x^{5} y^{9}} \) using the properties of rational numbers, follow these ordered steps: ### **Step 1: Factor Inside the Cube Root** Break down each component inside the cube root into factors that include perfect cubes. - **Factor 875:** \( 875 = 125 \times 7 \) (Since \(125 = 5^3\) is a perfect cube) - **Factor \(x^5\):** \( x^5 = x^3 \times x^2 \) (Here, \(x^3\) is a perfect cube) - **Factor \(y^9\):** \( y^9 = (y^3)^3 \) (Since \(y^3\) raised to the power of 3 is a perfect cube) The expression now looks like: \[ \sqrt[3]{125 \times 7 \times x^3 \times x^2 \times (y^3)^3} \] ### **Step 2: Apply the Property of Cube Roots** Use the property \( \sqrt[3]{a \times b \times c} = \sqrt[3]{a} \times \sqrt[3]{b} \times \sqrt[3]{c} \) to separate the factors. \[ \sqrt[3]{125} \times \sqrt[3]{7} \times \sqrt[3]{x^3} \times \sqrt[3]{x^2} \times \sqrt[3]{(y^3)^3} \] ### **Step 3: Simplify Each Cube Root** Evaluate the cube roots that are perfect cubes. - \( \sqrt[3]{125} = 5 \) - \( \sqrt[3]{x^3} = x \) - \( \sqrt[3]{(y^3)^3} = y^3 \) So, the expression simplifies to: \[ 5 \times \sqrt[3]{7} \times x \times \sqrt[3]{x^2} \times y^3 \] ### **Step 4: Combine Like Terms and Constants** Multiply the constants and like terms together to finalize the simplified expression. \[ 5x y^3 \sqrt[3]{7 x^2} \] ### **Final Simplified Expression** \[ 5x y^{3} \sqrt[3]{7 x^{2}} \] This is the simplified form of the original expression using the properties of rational numbers.