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Simplify this radical:

[tex]\[ \sqrt{84} \][/tex]

A. [tex]\( 2 \sqrt{21} \)[/tex]

B. [tex]\( 2 \sqrt{42} \)[/tex]

C. [tex]\( 4 \sqrt{21} \)[/tex]

D. [tex]\( 4 \sqrt{42} \)[/tex]

Sagot :

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]