Verified Answer :
- To simplify \(\frac{\sqrt{7}}{\sqrt{11}}\), rationalize the denominator by multiplying the numerator and denominator by \(\sqrt{11}\).
- Apply the multiplication rule \(\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}\) to the numerator to get \(\sqrt{77}\).
- Simplify the denominator by calculating \(\sqrt{11} \times \sqrt{11} = 11\).
- The expression in simplest radical form is \(\boxed{\frac{\sqrt{77}}{11}}\).
Explanation
- Analyzing the Problem
We need to express the fraction \(\frac{\sqrt{7}}{\sqrt{11}}\) in simplest radical form. In mathematics, this means rationalizing the denominator, which involves manipulating the fraction so that the denominator contains only rational numbers. The given expression has a square root in the denominator, \(\sqrt{11}\), which is an irrational number. Our goal is to eliminate this by finding an equivalent expression with a rational denominator.
- Rationalizing the Denominator
To rationalize the denominator of \(\frac{\sqrt{7}}{\sqrt{11}}\), we multiply both the numerator and the denominator by \(\sqrt{11}\). This is based on the property that multiplying by \(\frac{\sqrt{11}}{\sqrt{11}}\) (which is equivalent to multiplying by \(1\)) does not change the value of the fraction. The calculation is as follows:
\[\frac{\sqrt{7}}{\sqrt{11}} \times \frac{\sqrt{11}}{\sqrt{11}} = \frac{\sqrt{7} \times \sqrt{11}}{\sqrt{11} \times \sqrt{11}}.\]
- Simplifying the Expression and Finalizing the Answer
Now, we simplify the numerator and the denominator. Using the product rule for radicals, \(\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}\), the numerator becomes \(\sqrt{7 \times 11} = \sqrt{77}\). The denominator becomes \(\sqrt{11} \times \sqrt{11} = 11\). Thus, the fraction becomes \(\frac{\sqrt{77}}{11}\). Since \(77 = 7 \times 11\) and neither factor is a perfect square, the expression \(\sqrt{77}\) cannot be simplified further. Therefore, the fraction \(\frac{\sqrt{77}}{11}\) is in its simplest radical form.
Examples
Rationalizing denominators is frequently used in engineering and physics when performing calculations involving physical quantities like resistance or impedance. For instance, if you are calculating the impedance of an electrical circuit where an expression results in a fraction with a square root in the denominator, rationalizing it makes the value much easier to plot, analyze, or compare with other measurements. This process ensures that final values are standard and easier to work with in technical documentation and real-world applications.