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: \[ \boxed{2\sqrt{1099} + 2\sqrt{11} - 5\sqrt{91}} \]