Sure! Let's solve the given expression step by step.
Given: [tex]\( 25 y^2 \sqrt[3]{64 x^5} \div \sqrt[3]{8 x^2} \)[/tex], where [tex]\( x \neq 0 \)[/tex].
### Step 1: Simplify the cube roots
First, we simplify [tex]\( \sqrt[3]{64 x^5} \)[/tex]:
[tex]\[ \sqrt[3]{64 x^5} \][/tex]
[tex]\[ 64 x^5 \text{ can be broken down into } (4^3 \cdot x^5) \][/tex]
[tex]\[ \sqrt[3]{64 x^5} = \sqrt[3]{4^3 \cdot x^5} \][/tex]
[tex]\[ = 4 \cdot x^{5/3} \][/tex]
Next, we simplify [tex]\( \sqrt[3]{8 x^2} \)[/tex]:
[tex]\[ \sqrt[3]{8 x^2} \][/tex]
[tex]\[ 8 x^2 \text{ can be broken down into } (2^3 \cdot x^2) \][/tex]
[tex]\[ \sqrt[3]{8 x^2} = \sqrt[3]{2^3 \cdot x^2} \][/tex]
[tex]\[ = 2 \cdot x^{2/3} \][/tex]
### Step 2: Divide the simplified cube roots
Now we divide the simplified expressions:
[tex]\[ \frac{4 \cdot x^{5/3}}{2 \cdot x^{2/3}} \][/tex]
[tex]\[ = \frac{4}{2} \cdot \frac{x^{5/3}}{x^{2/3}} \][/tex]
[tex]\[ = 2 \cdot x^{(5/3 - 2/3)} \][/tex]
[tex]\[ = 2 \cdot x^{3/3} \][/tex]
[tex]\[ = 2 \cdot x \][/tex]
### Step 3: Combine with the rest of the expression
The original expression now becomes:
[tex]\[ 25 y^2 \cdot 2 \cdot x \][/tex]
[tex]\[ = 50 x y^2 \][/tex]
Therefore, the simplified expression is:
[tex]\[ \boxed{50 x y^2} \][/tex]
