Drag each tile to the correct box.
Order the simplification steps of the expression below using the properties of rational

Ask by Burgess Flynn. in the United States

Jan 23,2025

Question

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

UpStudy AI · Accepted Answer

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.
Simplify the expression $$ \sqrt[3]{875 x^{5} y^{9}} $$ by factoring and applying cube root properties: 1. **Factor Inside the Cube Root:** - $$ 875 = 125 \times 7 $$ - $$ x^5 = x^3 \times x^2 $$ - $$ y^9 = (y^3)^3 $$ 2. **Apply Cube Root Properties:** $$ \sqrt[3]{125} \times \sqrt[3]{7} \times \sqrt[3]{x^3} \times \sqrt[3]{x^2} \times \sqrt[3]{y^9} $$ 3. **Simplify Perfect Cubes:** - $$ \sqrt[3]{125} = 5 $$ - $$ \sqrt[3]{x^3} = x $$ - $$ \sqrt[3]{y^9} = y^3 $$ 4. **Combine Terms:** $$ 5 \times x \times y^3 \times \sqrt[3]{7 x^2} = 5x y^{3} \sqrt[3]{7 x^{2}} $$ **Final Simplified Expression:** $$ 5x y^{3} \sqrt[3]{7 x^{2}} $$

Drag each tile to the correct box.
Order the simplification steps of the expression below using the properties of rational

Upstudy AI Solution

Answer

Solution

Step 1: Factor Inside the Cube Root

Step 2: Apply the Property of Cube Roots

Step 3: Simplify Each Cube Root

Step 4: Combine Like Terms and Constants

Final Simplified Expression

Beyond the Answer

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