Sure! Let's go through the steps to understand the expression for [tex]\( y \)[/tex] given as:
[tex]\[ y = 3 x^2 \sqrt{2 x^3 - 3} \][/tex]
### Step 1: Identify the Components
Firstly, let's identify the components involved in the expression:
- The term [tex]\( 3 \)[/tex] is a constant multiplier.
- The term [tex]\( x^2 \)[/tex] involves [tex]\( x \)[/tex] raised to the power of 2.
- The term [tex]\( \sqrt{2 x^3 - 3} \)[/tex] is a square root function involving another expression inside it.
### Step 2: Understand the Nested Expression
Let's look inside the square root specifically:
[tex]\[ \sqrt{2 x^3 - 3} \][/tex]
Here, [tex]\( 2 x^3 - 3 \)[/tex] is a polynomial expression where [tex]\( 2 x^3 \)[/tex] refers to [tex]\( 2 \)[/tex] times [tex]\( x \)[/tex] raised to the power of 3, and then subtracting 3 from it.
### Step 3: Combine All Components
Next, we combine the individual components together:
- Multiply [tex]\( x^2 \)[/tex] by 3, giving [tex]\( 3 x^2 \)[/tex].
- Multiply this result by the square root function [tex]\( \sqrt{2 x^3 - 3} \)[/tex].
This results in the final expression:
[tex]\[ y = 3 x^2 \sqrt{2 x^3 - 3} \][/tex]
### Step 4: Interpretation
The expression [tex]\( y = 3 x^2 \sqrt{2 x^3 - 3} \)[/tex] combines polynomial and radical components in a way that depends on the value of [tex]\( x \)[/tex]. For positive values of [tex]\( x \)[/tex], the term inside the square root should be non-negative for [tex]\( y \)[/tex] to be real-valued.
Hence, we've examined and detailed the components and formation of the given expression [tex]\( y = 3 x^2 \sqrt{2 x^3 - 3} \)[/tex].
