Answer :
To express \(\sqrt{98}\) in its simplest radical form, follow these steps:
1. Find the prime factorization of 98:
- First, divide 98 by the smallest prime number, which is 2:
[tex]\[ 98 \div 2 = 49 \][/tex]
- Next, factor 49, which is \(7 \times 7\):
Thus, the prime factorization of 98 is:
[tex]\[ 98 = 2 \times 7 \times 7 \][/tex]
2. Group the factors into pairs of squares when possible:
- Notice that \(98 = 2 \times 7^2\)
3. Simplify the square root by separating the product of the square root of the square number and the remaining number:
- Using the properties of square roots:
[tex]\[ \sqrt{98} = \sqrt{2 \times 7^2} \][/tex]
- This can be broken down using the square root of a product rule:
[tex]\[ \sqrt{2 \times 7^2} = \sqrt{2} \times \sqrt{7^2} \][/tex]
- Since \(\sqrt{7^2} = 7\), the expression simplifies to:
[tex]\[ \sqrt{2} \times 7 = 7 \sqrt{2} \][/tex]
Therefore, the simplest radical form of \(\sqrt{98}\) is:
[tex]\[ \boxed{7 \sqrt{2}} \][/tex]
1. Find the prime factorization of 98:
- First, divide 98 by the smallest prime number, which is 2:
[tex]\[ 98 \div 2 = 49 \][/tex]
- Next, factor 49, which is \(7 \times 7\):
Thus, the prime factorization of 98 is:
[tex]\[ 98 = 2 \times 7 \times 7 \][/tex]
2. Group the factors into pairs of squares when possible:
- Notice that \(98 = 2 \times 7^2\)
3. Simplify the square root by separating the product of the square root of the square number and the remaining number:
- Using the properties of square roots:
[tex]\[ \sqrt{98} = \sqrt{2 \times 7^2} \][/tex]
- This can be broken down using the square root of a product rule:
[tex]\[ \sqrt{2 \times 7^2} = \sqrt{2} \times \sqrt{7^2} \][/tex]
- Since \(\sqrt{7^2} = 7\), the expression simplifies to:
[tex]\[ \sqrt{2} \times 7 = 7 \sqrt{2} \][/tex]
Therefore, the simplest radical form of \(\sqrt{98}\) is:
[tex]\[ \boxed{7 \sqrt{2}} \][/tex]
