To solve the equation \( y^3 = \frac{38y}{3} = \frac{3}{8} \), we will take each part step by step.
First, let's handle the equation \( y^3 = \frac{3}{8} \):
- Set the first part: \[ y^3 = \frac{3}{8} \] To find \( y \), take the cube root of both sides: \[ y = \sqrt[3]{\frac{3}{8}} = \frac{\sqrt[3]{3}}{\sqrt[3]{8}} = \frac{\sqrt[3]{3}}{2} \]
Now we can summarize this step:
\[ y = \frac{\sqrt[3]{3}}{2} \]
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Set the second part: Now let’s also consider the equation \( \frac{38y}{3} = \frac{3}{8} \):
- Multiply both sides by 3 to eliminate the fraction: \[ 38y = \frac{3 \cdot 3}{8} \quad \Rightarrow \quad 38y = \frac{9}{8} \]
- Now, divide by 38: \[ y = \frac{9}{8 \cdot 38} = \frac{9}{304} \]
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Combining results: So, we have two different solutions from both parts:
- From the first part: \( y = \frac{\sqrt[3]{3}}{2} \)
- From the second part: \( y = \frac{9}{304} \)
The final solutions for \( y \) are: \[ y = \frac{\sqrt[3]{3}}{2} \quad \text{or} \quad y = \frac{9}{304} \]
If you need both values as answers, they are: \[ y = \frac{\sqrt[3]{3}}{2} , \quad y = \frac{9}{304} \]