Answer :
Let's determine which of the given expressions are equivalent to [tex]\(\sqrt{252}\)[/tex]:
1. [tex]\(252^{\frac{1}{2}}\)[/tex]:
- We know that taking the square root of a number is equivalent to raising that number to the power of [tex]\(\frac{1}{2}\)[/tex].
- So, [tex]\(252^{\frac{1}{2}}\)[/tex] is indeed equal to [tex]\(\sqrt{252}\)[/tex].
Therefore, [tex]\(252^{\frac{1}{2}}\)[/tex] is equivalent to [tex]\(\sqrt{252}\)[/tex].
2. [tex]\(126^{\frac{1}{2}}\)[/tex]:
- Here, we are taking the square root of 126, which is not the same as taking the square root of 252.
- [tex]\(\sqrt{126}\)[/tex] cannot be simplified to [tex]\(\sqrt{252}\)[/tex].
Therefore, [tex]\(126^{\frac{1}{2}}\)[/tex] is not equivalent to [tex]\(\sqrt{252}\)[/tex].
3. [tex]\(7 \sqrt{6}\)[/tex]:
- To determine if this is equivalent to [tex]\(\sqrt{252}\)[/tex], let's see if we can simplify or transform it to match [tex]\(\sqrt{252}\)[/tex].
- Given: [tex]\( \sqrt{252} \)[/tex].
- Notice [tex]\( \sqrt{252} = 252^{\frac{1}{2}} \)[/tex].
- Let's factorize 252: [tex]\( 252 = 2^2 \times 3^2 \times 7 \)[/tex].
- Splitting the factors inside the square root: [tex]\( \sqrt{252} = \sqrt{(2^2 \times 3^2 \times 7)} = \sqrt{(2^2 \times 3^2) \times 7} = \sqrt{36 \times 7} = \sqrt{36} \times \sqrt{7} = 6 \sqrt{7} \)[/tex].
- So, [tex]\( 7 \sqrt{6} \neq \sqrt{252} \)[/tex].
Therefore, [tex]\(7 \sqrt{6}\)[/tex] is not equivalent to [tex]\(\sqrt{252}\)[/tex].
4. [tex]\(18 \sqrt{7}\)[/tex]:
- To determine if this is equivalent to [tex]\(\sqrt{252}\)[/tex], let's compare it with [tex]\(\sqrt{252}\)[/tex].
- From the earlier factorization, we found [tex]\( \sqrt{252} = 6 \sqrt{7} \)[/tex].
- Clearly, [tex]\( 18 \sqrt{7} \neq 6 \sqrt{7} \)[/tex].
Therefore, [tex]\(18 \sqrt{7}\)[/tex] is not equivalent to [tex]\(\sqrt{252}\)[/tex].
5. [tex]\(6 \sqrt{7}\)[/tex]:
- From our earlier factorization work, we simplified [tex]\(\sqrt{252}\)[/tex] as [tex]\(\sqrt{36 \times 7} = 6 \sqrt{7}\)[/tex].
- This matches perfectly.
Therefore, [tex]\(6 \sqrt{7}\)[/tex] is equivalent to [tex]\(\sqrt{252}\)[/tex].
So, the expressions that are equivalent to [tex]\(\sqrt{252}\)[/tex] are:
[tex]\[ 252^{\frac{1}{2}} \quad \text{and} \quad 6 \sqrt{7} \][/tex]
1. [tex]\(252^{\frac{1}{2}}\)[/tex]:
- We know that taking the square root of a number is equivalent to raising that number to the power of [tex]\(\frac{1}{2}\)[/tex].
- So, [tex]\(252^{\frac{1}{2}}\)[/tex] is indeed equal to [tex]\(\sqrt{252}\)[/tex].
Therefore, [tex]\(252^{\frac{1}{2}}\)[/tex] is equivalent to [tex]\(\sqrt{252}\)[/tex].
2. [tex]\(126^{\frac{1}{2}}\)[/tex]:
- Here, we are taking the square root of 126, which is not the same as taking the square root of 252.
- [tex]\(\sqrt{126}\)[/tex] cannot be simplified to [tex]\(\sqrt{252}\)[/tex].
Therefore, [tex]\(126^{\frac{1}{2}}\)[/tex] is not equivalent to [tex]\(\sqrt{252}\)[/tex].
3. [tex]\(7 \sqrt{6}\)[/tex]:
- To determine if this is equivalent to [tex]\(\sqrt{252}\)[/tex], let's see if we can simplify or transform it to match [tex]\(\sqrt{252}\)[/tex].
- Given: [tex]\( \sqrt{252} \)[/tex].
- Notice [tex]\( \sqrt{252} = 252^{\frac{1}{2}} \)[/tex].
- Let's factorize 252: [tex]\( 252 = 2^2 \times 3^2 \times 7 \)[/tex].
- Splitting the factors inside the square root: [tex]\( \sqrt{252} = \sqrt{(2^2 \times 3^2 \times 7)} = \sqrt{(2^2 \times 3^2) \times 7} = \sqrt{36 \times 7} = \sqrt{36} \times \sqrt{7} = 6 \sqrt{7} \)[/tex].
- So, [tex]\( 7 \sqrt{6} \neq \sqrt{252} \)[/tex].
Therefore, [tex]\(7 \sqrt{6}\)[/tex] is not equivalent to [tex]\(\sqrt{252}\)[/tex].
4. [tex]\(18 \sqrt{7}\)[/tex]:
- To determine if this is equivalent to [tex]\(\sqrt{252}\)[/tex], let's compare it with [tex]\(\sqrt{252}\)[/tex].
- From the earlier factorization, we found [tex]\( \sqrt{252} = 6 \sqrt{7} \)[/tex].
- Clearly, [tex]\( 18 \sqrt{7} \neq 6 \sqrt{7} \)[/tex].
Therefore, [tex]\(18 \sqrt{7}\)[/tex] is not equivalent to [tex]\(\sqrt{252}\)[/tex].
5. [tex]\(6 \sqrt{7}\)[/tex]:
- From our earlier factorization work, we simplified [tex]\(\sqrt{252}\)[/tex] as [tex]\(\sqrt{36 \times 7} = 6 \sqrt{7}\)[/tex].
- This matches perfectly.
Therefore, [tex]\(6 \sqrt{7}\)[/tex] is equivalent to [tex]\(\sqrt{252}\)[/tex].
So, the expressions that are equivalent to [tex]\(\sqrt{252}\)[/tex] are:
[tex]\[ 252^{\frac{1}{2}} \quad \text{and} \quad 6 \sqrt{7} \][/tex]
