What is $\frac{\sqrt{7}}{\sqrt{11}}$ in simplest radical form?
To express \(\frac{\sqrt{7}}{\sqrt{11}}\) in simplest radical form, we must rationalize the denominator. This means we want to eliminate the radical sign from the denominator, leaving it as a rational number. We achieve this by multiplying both the numerator and the denominator by \(\sqrt{11}\).
Multiplying the fraction by \(\frac{\sqrt{11}}{\sqrt{11}}\) does not change its value because \(\frac{\sqrt{11}}{\sqrt{11}} = 1\). This gives us:
\[\frac{\sqrt{7}}{\sqrt{11}} \cdot \frac{\sqrt{11}}{\sqrt{11}} = \frac{\sqrt{7} \cdot \sqrt{11}}{\sqrt{11} \cdot \sqrt{11}}.\]
Using the property of square roots that \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\), the numerator becomes \(\sqrt{7 \cdot 11} = \sqrt{77}\). In the denominator, \(\sqrt{11} \cdot \sqrt{11} = 11\). Thus, the expression becomes:
\[\frac{\sqrt{77}}{11}.\] Since 77 has no square factors other than 1 (its prime factorization is \(7 \times 11\)), the radical \(\sqrt{77}\) cannot be simplified further. Therefore, this is the simplest radical form of the expression.
In engineering or architecture, you might often encounter expressions with radicals in the denominator when calculating ratios, slopes, or dimensions of right-angled triangles using the Pythagorean theorem. Rationalizing these expressions is a standard practice because it makes it much easier to compare values, perform further calculations, or input them into software programs that prefer simplified forms. It transforms an irrational denominator into a rational one, which simplifies mental math and reduces potential errors in complex formulas.