Question

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

Answer 1

To combine the radicals \( 4\sqrt{396} + \sqrt{44} - 2\sqrt{275} \), we first simplify each radical term individually.

1. **Simplifying \( \sqrt{396} \)**:
\[
396 = 4 \times 99 = 4 \times 9 \times 11 = 36 \times 11
\]
Thus,
\[
\sqrt{396} = \sqrt{4 \times 99} = \sqrt{4} \times \sqrt{99} = 2\sqrt{99}
\]
Now we can compute \( 4\sqrt{396} \):
\[
4\sqrt{396} = 4 \times 2\sqrt{99} = 8\sqrt{99}
\]

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

3. **Simplifying \( \sqrt{275} \)**:
\[
275 = 25 \times 11 = 5^2 \times 11
\]
Thus,
\[
\sqrt{275} = \sqrt{25 \times 11} = \sqrt{25} \times \sqrt{11} = 5\sqrt{11}
\]
Now we can compute \( 2\sqrt{275} \):
\[
2\sqrt{275} = 2 \times 5\sqrt{11} = 10\sqrt{11}
\]

Now we substitute back into the original expression:

\[
4\sqrt{396} + \sqrt{44} - 2\sqrt{275} = 8\sqrt{99} + 2\sqrt{11} - 10\sqrt{11}
\]

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

\[
8\sqrt{99} + (2\sqrt{11} - 10\sqrt{11}) = 8\sqrt{99} - 8\sqrt{11}
\]

Finally, we factor out the common term:

\[
8\sqrt{99} - 8\sqrt{11} = 8(\sqrt{99} - \sqrt{11})
\]

Thus, the final result in simplest radical form is:

\[
\boxed{8(\sqrt{99} - \sqrt{11})}
\]