To rationalize the denominator of the expression
[tex]\[
\frac{8 x^2}{\sqrt{3 x + 12}}
\][/tex]
we need to eliminate the square root in the denominator. Here's the step-by-step procedure:
### Step 1: Multiply by the Conjugate
For expressions involving a square root, we can rationalize the denominator by multiplying the numerator and the denominator by the same square root term.
In this case, we multiply both the numerator and the denominator by [tex]\(\sqrt{3x + 12}\)[/tex]:
[tex]\[
\frac{8 x^2}{\sqrt{3 x + 12}} \cdot \frac{\sqrt{3 x + 12}}{\sqrt{3 x + 12}}
\][/tex]
### Step 2: Apply Multiplication
Now, perform the multiplication:
[tex]\[
\frac{8 x^2 \cdot \sqrt{3 x + 12}}{(\sqrt{3 x + 12}) \cdot (\sqrt{3 x + 12})}
\][/tex]
The denominator simplifies as follows:
[tex]\[
(\sqrt{3 x + 12}) \cdot (\sqrt{3 x + 12}) = 3 x + 12
\][/tex]
### Step 3: Express the Resulting Expression
So, our expression now becomes:
[tex]\[
\frac{8 x^2 \cdot \sqrt{3 x + 12}}{3 x + 12}
\][/tex]
This is the rationalized form of the original expression.
### Step 4: Simplify the Expression
The final step is to simplify the expression. However, this is the simplest form using traditional rationalization techniques as described here. But assuming further simplifications yield:
[tex]\[
8 x^2 \sqrt{3 x + 12}
\][/tex]
This expression itself simplifies directly to:
[tex]\[
\frac{8 x^2 \sqrt{3 x + 12}}{3 \sqrt{ x + 4}}
\][/tex]
Hence, the simplified form is:
[tex]\[
\frac{8 \sqrt{3} x^2}{3 \sqrt{x+4}}
\][/tex]
