To express \(\sqrt{112}\) in simplest radical form, follow these steps:
1. Find the prime factorization of 112:
- Start by dividing by the smallest prime number, which is 2:
[tex]\[
112 \div 2 = 56
\][/tex]
- Continue dividing 56 by 2:
[tex]\[
56 \div 2 = 28
\][/tex]
- Continue dividing 28 by 2:
[tex]\[
28 \div 2 = 14
\][/tex]
- Continue dividing 14 by 2:
[tex]\[
14 \div 2 = 7
\][/tex]
- Finally, 7 is a prime number.
Therefore, the prime factorization of 112 is:
[tex]\[
112 = 2^4 \times 7
\][/tex]
2. Express the square root of 112 using its prime factors:
[tex]\[
\sqrt{112} = \sqrt{2^4 \times 7}
\][/tex]
3. Simplify the square root by taking the square root of the factors:
- The square root of \(2^4\) is \(2^2\), since \((2^2)^2 = 2^4\).
- The square root of 7 remains \(\sqrt{7}\), as 7 is a prime number and cannot be simplified further.
Expressing this simplification:
[tex]\[
\sqrt{112} = \sqrt{2^4 \times 7} = \sqrt{(2^2)^2 \times 7} = 2^2 \times \sqrt{7}
\][/tex]
4. Calculate the final simplified form:
[tex]\[
2^2 = 4
\][/tex]
Therefore:
[tex]\[
\sqrt{112} = 4 \sqrt{7}
\][/tex]
Thus, the simplest radical form of \(\sqrt{112}\) is:
[tex]\[
\boxed{4 \sqrt{7}}
\][/tex]
