Question

Multiply. Assume c is greater than or equal to zero, and write your answer in simplest form.
√22c x √14c

Answer 1

To multiply the expressions \(\sqrt{22c}\) and \(\sqrt{14c}\), we can use the property of square roots that states \(\sqrt{a} \times \sqrt{b} = \sqrt{ab}\).

So, we have:

\[
\sqrt{22c} \times \sqrt{14c} = \sqrt{(22c)(14c)}
\]

Now, let's multiply the terms inside the square root:

\[
(22c)(14c) = 22 \times 14 \times c \times c = 308c^2
\]

Next, we can write this as:

\[
\sqrt{308c^2}
\]

Since \(c\) is non-negative, we can take the square root of \(c^2\):

\[
\sqrt{308} \times \sqrt{c^2} = \sqrt{308} \times c
\]

Now, we need to simplify \(\sqrt{308}\). We can factor \(308\) into its prime factors:

\[
308 = 4 \times 77 = 4 \times 7 \times 11 = 2^2 \times 7 \times 11
\]

The square root can be simplified as follows:

\[
\sqrt{308} = \sqrt{4 \times 77} = \sqrt{4} \times \sqrt{77} = 2\sqrt{77}
\]

Finally, we combine this back with \(c\):

\[
\sqrt{22c} \times \sqrt{14c} = 2\sqrt{77} \times c = 2c\sqrt{77}
\]

Thus, the final answer in simplest form is:

\[
\boxed{2c\sqrt{77}}
\]