Sure! Let's work through the problem step by step.
### Problem Statement:
Find the expression for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex].
### Solution:
We need to find the expression [tex]\( y \)[/tex] given the equation:
[tex]\[ y = 3 x^2 \sqrt{2 x^3 - 3} \][/tex]
Here’s a detailed step-by-step breakdown:
1. Identify the components of the equation:
- [tex]\( 3x^2 \)[/tex]: This is a polynomial term where [tex]\( x \)[/tex] is raised to the power of 2 and then multiplied by 3.
- [tex]\( \sqrt{2 x^3 - 3} \)[/tex]: This is a square root term where the expression inside the square root is [tex]\( 2 x^3 - 3 \)[/tex].
2. Understand the relationship:
- The polynomial term [tex]\( 3 x^2 \)[/tex] is being multiplied by the square root term [tex]\( \sqrt{2 x^3 - 3} \)[/tex].
- This means the final expression for [tex]\( y \)[/tex] is the product of these two terms.
3. Expressing in terms of [tex]\( x \)[/tex]:
- The expression [tex]\( 2 x^3 - 3 \)[/tex] inside the square root helps determine how the overall function behaves in terms of [tex]\( x \)[/tex].
- When calculating [tex]\( y \)[/tex], first compute [tex]\( 2 x^3 - 3 \)[/tex], then take the square root of the result, and finally multiply by [tex]\( 3 x^2 \)[/tex].
Thus, the expression for [tex]\( y \)[/tex] simplifies directly to:
[tex]\[ y = 3 x^2 \sqrt{2 x^3 - 3} \][/tex]
No further simplification is possible in this case as it already appears in its simplest form.
### Conclusion:
The expression for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex] is given by:
[tex]\[ y = 3 x^2 \sqrt{2 x^3 - 3} \][/tex]
This is the final form of the equation and provides a clear relationship between [tex]\( y \)[/tex] and [tex]\( x \)[/tex].
