To determine which expression the square root of 162 is equivalent to, let's simplify [tex]\(\sqrt{162}\)[/tex] step by step.
### Step 1: Factorize 162
First, we need to find the prime factors of 162.
- [tex]\(162\)[/tex] is even, so we can divide by [tex]\(2\)[/tex]:
[tex]\[
162 \div 2 = 81
\][/tex]
- Next, [tex]\(81\)[/tex] is divisible by [tex]\(3\)[/tex] (since the sum of the digits [tex]\(8 + 1 = 9\)[/tex] and [tex]\(9\)[/tex] is divisible by [tex]\(3\)[/tex]):
[tex]\[
81 \div 3 = 27
\][/tex]
- Continuing, [tex]\(27\)[/tex] is also divisible by [tex]\(3\)[/tex]:
[tex]\[
27 \div 3 = 9
\][/tex]
- Again, [tex]\(9\)[/tex] is divisible by [tex]\(3\)[/tex]:
[tex]\[
9 \div 3 = 3
\][/tex]
- Finally, [tex]\(3\)[/tex] is a prime number.
So the prime factorization of [tex]\(162\)[/tex] is:
[tex]\[
162 = 2 \times 3 \times 3 \times 3 \times 3 = 2 \times 3^4
\][/tex]
### Step 2: Simplify the square root
Now we break down the expression within the square root using the prime factors:
[tex]\[
\sqrt{162} = \sqrt{2 \times 3^4}
\][/tex]
Since the square root of a product is the product of the square roots:
[tex]\[
\sqrt{2 \times 3^4} = \sqrt{2} \times \sqrt{3^4}
\][/tex]
### Step 3: Simplify further using powers
We know that:
[tex]\[
\sqrt{3^4} = 3^2 = 9
\][/tex]
### Final Simplification:
Thus:
[tex]\[
\sqrt{162} = \sqrt{2} \times 9 = 9\sqrt{2}
\][/tex]
### Conclusion:
The expression [tex]\(\sqrt{162}\)[/tex] is equivalent to:
[tex]\[
\boxed{9\sqrt{2}}
\][/tex]
Therefore, the correct option is:
(C) [tex]\(9 \sqrt{2}\)[/tex]
