To rationalize the denominator of [tex]\(\frac{2 \sqrt{10}}{3 \sqrt{11}}\)[/tex], you need to eliminate the square root in the denominator. Here is the step-by-step process to do so:
1. Understand the problem: We need to rationalize the denominator of [tex]\(\frac{2 \sqrt{10}}{3 \sqrt{11}}\)[/tex]. This means we must eliminate the square root that appears in the denominator.
2. Identify the term in the denominator that requires rationalization: In this case, the term is [tex]\(3 \sqrt{11}\)[/tex]. To get rid of the square root, we multiply the denominator (and the numerator) by [tex]\(\sqrt{11}\)[/tex].
3. Multiply the expression by a fraction equal to 1 that contains [tex]\(\sqrt{11}\)[/tex] in both the numerator and the denominator: The appropriate fraction used for rationalization would be [tex]\(\frac{\sqrt{11}}{\sqrt{11}}\)[/tex].
4. Perform the multiplication:
[tex]\[
\frac{2 \sqrt{10}}{3 \sqrt{11}} \times \frac{\sqrt{11}}{\sqrt{11}} = \frac{2 \sqrt{10} \cdot \sqrt{11}}{3 \sqrt{11} \cdot \sqrt{11}}
\][/tex]
5. Simplify the expression: Use the property of square roots that [tex]\(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\)[/tex]:
[tex]\[
= \frac{2 \sqrt{110}}{3 \cdot 11}
\][/tex]
6. Simplify the denominator:
[tex]\[
= \frac{2 \sqrt{110}}{33}
\][/tex]
Thus, the fraction by which you need to multiply [tex]\(\frac{2 \sqrt{10}}{3 \sqrt{11}}\)[/tex] in order to rationalize the denominator is [tex]\(\boxed{\frac{\sqrt{11}}{\sqrt{11}}}\)[/tex].
