Question

Combine the radicals using addition and subtraction. Give the answer in simplest radical form.
4√396 + √44 - 2√275

Answer 1

To combine the radicals in the expression \( 4\sqrt{396} + \sqrt{44} - 2\sqrt{275} \), we will simplify each radical first:

1. **Simplifying \( \sqrt{396} \)**:

To simplify \( \sqrt{396} \):

- Break down \( 396 \) into its prime factors:
\[
396 = 4 \times 99 = 4 \times 9 \times 11 = 36 \times 11.
\]
- Thus, \( \sqrt{396} = \sqrt{36 \times 11} = \sqrt{36} \times \sqrt{11} = 6\sqrt{11} \).
- Therefore, \( 4\sqrt{396} = 4 \times 6\sqrt{11} = 24\sqrt{11} \).

2. **Simplifying \( \sqrt{44} \)**:

To simplify \( \sqrt{44} \):
- Break down \( 44 \):
\[
44 = 4 \times 11.
\]
- Therefore, \( \sqrt{44} = \sqrt{4 \times 11} = \sqrt{4} \times \sqrt{11} = 2\sqrt{11} \).

3. **Simplifying \( \sqrt{275} \)**:

To simplify \( \sqrt{275} \):
- Break down \( 275 \):
\[
275 = 25 \times 11.
\]
- Therefore, \( \sqrt{275} = \sqrt{25 \times 11} = \sqrt{25} \times \sqrt{11} = 5\sqrt{11} \).
- Thus, \( -2\sqrt{275} = -2 \times 5\sqrt{11} = -10\sqrt{11} \).

Now, we can rewrite the entire expression:

\[
4\sqrt{396} + \sqrt{44} - 2\sqrt{275} = 24\sqrt{11} + 2\sqrt{11} - 10\sqrt{11}.
\]

Next, combine the terms involving \( \sqrt{11} \):

\[
(24 + 2 - 10)\sqrt{11} = 16\sqrt{11}.
\]

Thus, the final simplified form of the expression is:

\[
\boxed{16\sqrt{11}}.
\]