To express [tex]\(\sqrt{162}\)[/tex] in its simplest radical form, we need to break down the number under the square root into its prime factors and simplify accordingly.
1. Factorize the number 162:
- First, divide by the smallest prime, 2:
[tex]\[
162 \div 2 = 81
\][/tex]
- Next, since 81 is still not prime, we continue factoring. Note that 81 is a power of 3:
[tex]\[
81 = 3 \times 27
\][/tex]
[tex]\[
27 = 3 \times 9
\][/tex]
[tex]\[
9 = 3 \times 3
\][/tex]
Thus, [tex]\(162\)[/tex] can be expressed as:
[tex]\[
162 = 2 \times 3 \times 3 \times 3 \times 3 = 2 \times 3^4
\][/tex]
2. Simplify the square root:
- We have:
[tex]\[
\sqrt{162} = \sqrt{2 \times 3^4}
\][/tex]
- We can separate the square root into two parts based on multiplication under the square root:
[tex]\[
\sqrt{2 \times 3^4} = \sqrt{2} \times \sqrt{3^4}
\][/tex]
- Recognize that [tex]\(\sqrt{3^4} = \sqrt{(3^2)^2} = 3^2 = 9\)[/tex]:
[tex]\[
\sqrt{2} \times 9
\][/tex]
3. Combine and simplify the expression:
- Multiplication yields:
[tex]\[
9\sqrt{2}
\][/tex]
Therefore, the simplest radical form of [tex]\(\sqrt{162}\)[/tex] is:
[tex]\[
9\sqrt{2}
\][/tex]
So, the simplest radical form includes a constant term 9 and a radical term [tex]\(\sqrt{2}\)[/tex].
