Question

Express in simplest radical form.
start fraction, square root of, 504, end square root, divided by, square root of, 2, end square root, end fraction
2
​

504
​

​

Answer
Attempt 1 out of 5

Answer 1

To simplify the expression \(\frac{\sqrt{504}}{\sqrt{2}}\), we can start by simplifying the square root in the numerator.

First, let's simplify \(\sqrt{504}\). We can factor \(504\) into its prime factors:

\[
504 = 2^3 \times 3^2 \times 7
\]

Now, we can find \(\sqrt{504}\):

\[
\sqrt{504} = \sqrt{2^3 \times 3^2 \times 7}
\]

Using the properties of square roots, we can take the square root of each factor:

\[
\sqrt{504} = \sqrt{2^3} \times \sqrt{3^2} \times \sqrt{7}
\]

Calculating each part:

\[
\sqrt{2^3} = \sqrt{4 \times 2} = 2\sqrt{2}
\]
\[
\sqrt{3^2} = 3
\]
\[
\sqrt{7} = \sqrt{7}
\]

So we have:

\[
\sqrt{504} = 2\sqrt{2} \cdot 3 \cdot \sqrt{7} = 6\sqrt{14}
\]

Now substituting this back into our original expression:

\[
\frac{\sqrt{504}}{\sqrt{2}} = \frac{6\sqrt{14}}{\sqrt{2}}
\]

We can simplify this further:

\[
\frac{6\sqrt{14}}{\sqrt{2}} = 6\frac{\sqrt{14}}{\sqrt{2}} = 6\sqrt{\frac{14}{2}} = 6\sqrt{7}
\]

Thus, the simplest radical form of the given expression is:

\[
\boxed{6\sqrt{7}}
\]