To determine which expressions are equivalent to the given expression [tex]\(\sqrt{252}\)[/tex], we need to evaluate each one and compare it to [tex]\(\sqrt{252}\)[/tex].
First, we calculate the value of [tex]\(\sqrt{252}\)[/tex]:
[tex]\[
\sqrt{252} \approx 15.874507866387544
\][/tex]
Next, we compare [tex]\(\sqrt{252}\)[/tex] with each of the given expressions:
1. [tex]\(7 \sqrt{6}\)[/tex]:
[tex]\[
7 \sqrt{6} \approx 7 \times 2.449489742783178 = 17.146428199482244
\][/tex]
This is not equal to [tex]\(\sqrt{252}\)[/tex].
2. [tex]\(126^{\frac{1}{2}}\)[/tex]:
[tex]\[
126^{\frac{1}{2}} = \sqrt{126} \approx 11.224972160321824
\][/tex]
This is not equal to [tex]\(\sqrt{252}\)[/tex].
3. [tex]\(6 \sqrt{7}\)[/tex]:
[tex]\[
6 \sqrt{7} \approx 6 \times 2.6457513110645906 = 15.874507866387544
\][/tex]
This is equal to [tex]\(\sqrt{252}\)[/tex].
4. [tex]\(18 \sqrt{7}\)[/tex]:
[tex]\[
18 \sqrt{7} \approx 18 \times 2.6457513110645906 = 47.62352359916263
\][/tex]
This is not equal to [tex]\(\sqrt{252}\)[/tex].
5. [tex]\(252^{\frac{1}{2}}\)[/tex]:
[tex]\[
252^{\frac{1}{2}} = \sqrt{252} \approx 15.874507866387544
\][/tex]
This is obviously equal to [tex]\(\sqrt{252}\)[/tex].
From the calculations, the expressions that are equivalent to [tex]\(\sqrt{252}\)[/tex] are:
[tex]\[
6 \sqrt{7} \quad \text{and} \quad 252^{\frac{1}{2}}
\][/tex]
Therefore, the correct answers are:
[tex]\[
\boxed{6 \sqrt{7}} \quad \text{and} \quad \boxed{252^{\frac{1}{2}}}
\][/tex]
