To express [tex]\(\sqrt{18}\)[/tex] in its simplest radical form, follow these steps:
1. Find the prime factors of 18:
- The number 18 can be factored into [tex]\(2 \times 3 \times 3\)[/tex]. So, the prime factors of 18 are 2 and 3.
2. Express 18 under the square root as a product of its prime factors:
- [tex]\(\sqrt{18} = \sqrt{2 \times 3 \times 3}\)[/tex].
3. Simplify the radical by grouping the repeated prime factors:
- Notice that 3 appears twice. We can take one 3 out of the square root. This is because [tex]\(\sqrt{3 \times 3} = 3\)[/tex].
4. Rewrite the expression:
- [tex]\(\sqrt{18} = \sqrt{2 \times 3 \times 3} = \sqrt{2 \times (3 \times 3)} = 3\sqrt{2}\)[/tex].
So, the simplest radical form of [tex]\(\sqrt{18}\)[/tex] is:
[tex]\[ 3\sqrt{2} \][/tex]
In summary:
- The prime factors of 18 are 2 and 3.
- The coefficient outside the radical is 3.
- The value inside the radical is 2.
- Therefore, the simplest radical form of [tex]\(\sqrt{18}\)[/tex] is [tex]\(3\sqrt{2}\)[/tex].
