To simplify the radical [tex]\( \sqrt{84} \)[/tex], follow these steps:
1. Find the prime factorization of 84:
- Begin by dividing 84 by the smallest prime number, which is 2.
[tex]\[
84 \div 2 = 42
\][/tex]
- Next, divide 42 by 2 again.
[tex]\[
42 \div 2 = 21
\][/tex]
- Finally, divide 21 by the next smallest prime number, which is 3.
[tex]\[
21 \div 3 = 7
\][/tex]
- Now we have 7, which is a prime number.
The prime factorization of 84 is:
[tex]\[
84 = 2^2 \times 3 \times 7
\][/tex]
2. Apply the property of square roots that allows us to simplify radical expressions:
- Recall that [tex]\(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\)[/tex].
3. Group the factors to identify perfect squares:
- Observe that [tex]\( \sqrt{84} = \sqrt{2^2 \times 3 \times 7} \)[/tex].
- Notice that [tex]\(2^2\)[/tex] is a perfect square.
4. Simplify the expression:
- Extract the square root of [tex]\(2^2\)[/tex], which is 2, outside the radical.
[tex]\[
\sqrt{84} = \sqrt{2^2 \times 3 \times 7} = \sqrt{2^2} \times \sqrt{3 \times 7} = 2 \times \sqrt{21}
\][/tex]
Thus, the simplified form of [tex]\( \sqrt{84} \)[/tex] is:
[tex]\[
2\sqrt{21}
\][/tex]
Therefore, the correct answer is:
[tex]\[
2 \sqrt{21}
\][/tex]
