Question

Combine the radicals using addition and subtraction. Give the answer in simplest radical form.

6√117 − 3√52 + 4√13

Answer 1

To combine the radicals \( 6\sqrt{117} - 3\sqrt{52} + 4\sqrt{13} \), we first simplify each radical where possible.

1. **Simplify \( \sqrt{117} \)**:
\[
\sqrt{117} = \sqrt{9 \times 13} = \sqrt{9} \cdot \sqrt{13} = 3\sqrt{13}
\]
Thus, \( 6\sqrt{117} = 6(3\sqrt{13}) = 18\sqrt{13} \).

2. **Simplify \( \sqrt{52} \)**:
\[
\sqrt{52} = \sqrt{4 \times 13} = \sqrt{4} \cdot \sqrt{13} = 2\sqrt{13}
\]
Thus, \( -3\sqrt{52} = -3(2\sqrt{13}) = -6\sqrt{13} \).

3. **Combine the terms**:
Now we can combine all the terms:
\[
18\sqrt{13} - 6\sqrt{13} + 4\sqrt{13}
\]
Combine like terms:
\[
(18 - 6 + 4)\sqrt{13} = 16\sqrt{13}
\]

The final answer in simplest radical form is:
\[
\boxed{16\sqrt{13}}
\]