What is the following quotient?
$\frac{\sqrt{96}}{\sqrt{8}}$
We are asked to evaluate the expression \(\frac{\sqrt{96}}{\sqrt{8}}\). The expression is a quotient involving two square roots, and we aim to simplify it to one of the provided options. The relevant property here is the quotient rule for radicals, which states that for any non-negative real numbers \(a\) and positive real numbers \(b\), \(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\).
To simplify the expression \(\frac{\sqrt{96}}{\sqrt{8}}\), we can combine the terms under a single square root using the quotient rule:
\[ \frac{\sqrt{96}}{\sqrt{8}} = \sqrt{\frac{96}{8}} \]
Performing the division inside the radical:
\[ \frac{96}{8} = 12 \]
Thus, the expression simplifies to \(\sqrt{12}\).
Next, we simplify \(\sqrt{12}\) by identifying the perfect square factors within the number 12. We can factor 12 as \(4 \times 3\):
\[ \sqrt{12} = \sqrt{4 \times 3} \]
Using the product rule for radicals, \(\sqrt{x \times y} = \sqrt{x} \times \sqrt{y}\), we have:
\[ \sqrt{12} = \sqrt{4} \times \sqrt{3} \]
Since \(\sqrt{4} = 2\), the expression becomes:
\[ 2 \sqrt{3} \]
Comparing this result to the given options, we see that it corresponds to the first choice.
We have successfully simplified the quotient \(\frac{\sqrt{96}}{\sqrt{8}}\) to \(2 \sqrt{3}\). The steps taken were to combine the radicals, perform the division, and factor out perfect squares to simplify the resulting radical. The final answer is \(\boxed{2 \sqrt{3}}\).
Understanding how to manipulate radicals is essential in many fields, including architecture and construction. For example, if you are designing a support structure where lengths involve square roots, simplifying expressions like \(\frac{\sqrt{96}}{\sqrt{8}}\) helps in calculating exact measurements efficiently without needing to compute complex decimal approximations until the very final step of a project. This ensures precision in engineering and design tasks.