To rationalize the denominator of the expression [tex]\(\frac{6 x^4}{\sqrt{7 x+12}}\)[/tex], we follow these steps:
1. Start with the given expression:
[tex]\[
\frac{6 x^4}{\sqrt{7 x + 12}}
\][/tex]
2. Multiply both the numerator and the denominator by the conjugate of the denominator, which in this case is [tex]\(\sqrt{7 x + 12}\)[/tex]. This step helps us eliminate the square root in the denominator:
[tex]\[
\frac{6 x^4}{\sqrt{7 x + 12}} \cdot \frac{\sqrt{7 x + 12}}{\sqrt{7 x + 12}}
\][/tex]
3. Simplify the expression:
The numerator becomes:
[tex]\[
6 x^4 \cdot \sqrt{7 x + 12}
\][/tex]
The denominator becomes:
[tex]\[
\sqrt{7 x + 12} \cdot \sqrt{7 x + 12} = 7 x + 12
\][/tex]
4. Putting it all together, we get:
[tex]\[
\frac{6 x^4 \cdot \sqrt{7 x + 12}}{7 x + 12}
\][/tex]
So, the rationalized form of the expression [tex]\(\frac{6 x^4}{\sqrt{7 x + 12}}\)[/tex] is:
[tex]\[
\frac{6 x^4 \sqrt{7 x + 12}}{7 x + 12}
\][/tex]
