Question

Combine the radicals using addition and subtraction. Give the answer in simplest radical form. 4396−−−√+44−−√−2275−−−√

Answer 1

To combine the radicals \( \sqrt{4396} + \sqrt{44} - \sqrt{2275} \), we first simplify each radical where possible.

1. **Simplify \( \sqrt{4396} \)**:
- We can factor \( 4396 \):
\[
4396 = 4 \times 1099
\]
- Thus,
\[
\sqrt{4396} = \sqrt{4 \times 1099} = \sqrt{4} \times \sqrt{1099} = 2\sqrt{1099}
\]

2. **Simplify \( \sqrt{44} \)**:
- We factor \( 44 \):
\[
44 = 4 \times 11
\]
- So,
\[
\sqrt{44} = \sqrt{4 \times 11} = \sqrt{4} \times \sqrt{11} = 2\sqrt{11}
\]

3. **Simplify \( \sqrt{2275} \)**:
- We can factor \( 2275 \):
\[
2275 = 25 \times 91 = 25 \times (7 \times 13)
\]
- Thus,
\[
\sqrt{2275} = \sqrt{25 \times 91} = \sqrt{25} \times \sqrt{91} = 5\sqrt{91}
\]

Now we can substitute these simplified forms back into the expression:
\[
\sqrt{4396} + \sqrt{44} - \sqrt{2275} = 2\sqrt{1099} + 2\sqrt{11} - 5\sqrt{91}
\]

At this stage, we see that \( 2\sqrt{1099} \), \( 2\sqrt{11} \), and \( 5\sqrt{91} \) cannot be combined further because they involve different numbers under the radicals.

Thus, the final answer in the simplest radical form is:
\[
\boxed{2\sqrt{1099} + 2\sqrt{11} - 5\sqrt{91}}
\]