Answer :
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]
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]
