To rationalize the denominator of the expression [tex]\(\frac{11x}{\sqrt{6x + 7}}\)[/tex], follow these steps:
1. Identify the expression:
We start with the fraction [tex]\(\frac{11x}{\sqrt{6x + 7}}\)[/tex].
2. Multiply numerator and denominator by the conjugate of the denominator:
In this case, the "conjugate" of [tex]\(\sqrt{6x + 7}\)[/tex] is itself, since it is already in a simple radicand form. We multiply both the numerator and the denominator by [tex]\(\sqrt{6x + 7}\)[/tex] to eliminate the square root in the denominator:
[tex]\[
\frac{11x}{\sqrt{6x + 7}} \times \frac{\sqrt{6x + 7}}{\sqrt{6x + 7}}
\][/tex]
3. Simplify the expression:
When we multiply the numerators and the denominators together, we get:
[tex]\[
\frac{11x \sqrt{6x + 7}}{\sqrt{6x + 7} \cdot \sqrt{6x + 7}}
\][/tex]
The denominator becomes:
[tex]\[
\sqrt{6x + 7} \cdot \sqrt{6x + 7} = (\sqrt{6x + 7})^2 = 6x + 7
\][/tex]
So the fraction simplifies to:
[tex]\[
\frac{11x \sqrt{6x + 7}}{6x + 7}
\][/tex]
In conclusion, the rationalized form of [tex]\(\frac{11x}{\sqrt{6x + 7}}\)[/tex] is:
[tex]\[
\frac{11x \sqrt{6x + 7}}{6x + 7}
\][/tex]
