Question

Combine the radicals using addition and subtraction give the answer in simplest radical form for radical 396 + radical 44 - 2 radical 275

Answer 1

To combine the radicals \(\sqrt{396} + \sqrt{44} - 2\sqrt{275}\), we first simplify each radical.

1. **Simplifying \(\sqrt{396}\)**:
\[
396 = 4 \times 99 = 4 \times 9 \times 11 = 36 \times 11
\]
Therefore,
\[
\sqrt{396} = \sqrt{36 \times 11} = \sqrt{36} \times \sqrt{11} = 6\sqrt{11}
\]

2. **Simplifying \(\sqrt{44}\)**:
\[
44 = 4 \times 11
\]
Therefore,
\[
\sqrt{44} = \sqrt{4 \times 11} = \sqrt{4} \times \sqrt{11} = 2\sqrt{11}
\]

3. **Simplifying \(\sqrt{275}\)**:
\[
275 = 25 \times 11
\]
Therefore,
\[
\sqrt{275} = \sqrt{25 \times 11} = \sqrt{25} \times \sqrt{11} = 5\sqrt{11}
\]
So,
\[
2\sqrt{275} = 2 \times 5\sqrt{11} = 10\sqrt{11}
\]

Now we can substitute the simplified radicals back into the expression:
\[
\sqrt{396} + \sqrt{44} - 2\sqrt{275} = 6\sqrt{11} + 2\sqrt{11} - 10\sqrt{11}
\]

Next, we combine like terms:
\[
(6 + 2 - 10)\sqrt{11} = -2\sqrt{11}
\]

Thus, the answer in simplest radical form is:
\[
\boxed{-2\sqrt{11}}
\]