Let's determine which of the given expressions are equivalent to [tex]\(\sqrt{252}\)[/tex].
1. Expression: [tex]\(\sqrt{252}\)[/tex]
- This is the given expression. Clearly, [tex]\(\sqrt{252}\)[/tex] is equivalent to itself.
2. Expression: [tex]\(18 \sqrt{7}\)[/tex]
- Let's simplify [tex]\(18 \sqrt{7}\)[/tex].
- [tex]\(\sqrt{252} \approx 15.8745\)[/tex]
- Calculate [tex]\(18 \sqrt{7}\)[/tex]:
[tex]\[
18 \sqrt{7} \approx 18 \times 2.6458 = 47.6244
\][/tex]
- This is not equivalent to [tex]\(\sqrt{252}\)[/tex].
3. Expression: [tex]\(252^{\frac{1}{2}}\)[/tex]
- By definition, [tex]\(252^{\frac{1}{2}}\)[/tex] is another way of writing [tex]\(\sqrt{252}\)[/tex].
- This is equivalent to [tex]\(\sqrt{252}\)[/tex].
4. Expression: [tex]\(6 \sqrt{7}\)[/tex]
- Simplify [tex]\(6 \sqrt{7}\)[/tex].
- Calculate [tex]\(6 \sqrt{7}\)[/tex]:
[tex]\[
6 \sqrt{7} \approx 6 \times 2.6458 = 15.8748
\][/tex]
- This is not exactly correct due to rounding but theoretically, the values are very close.
5. Expression: [tex]\(7 \sqrt{6}\)[/tex]
- Simplify [tex]\(7 \sqrt{6}\)[/tex].
- Calculate [tex]\(7 \sqrt{6}\)[/tex]:
[tex]\[
7 \sqrt{6} \approx 7 \times 2.4495 = 17.1465
\][/tex]
- This is not equivalent to [tex]\(\sqrt{252}\)[/tex].
6. Expression: [tex]\(126^{\frac{1}{2}}\)[/tex]
- By definition, [tex]\(126^{\frac{1}{2}}\)[/tex] is another way of writing [tex]\(\sqrt{126}\)[/tex].
- Calculate [tex]\(\sqrt{126}\)[/tex]:
[tex]\[
\sqrt{126} \approx 11.2250
\][/tex]
- This is not equivalent to [tex]\(\sqrt{252}\)[/tex].
To conclude, the expressions equivalent to [tex]\(\sqrt{252}\)[/tex] are:
- [tex]\(252^{\frac{1}{2}}\)[/tex]
- [tex]\(6 \sqrt{7}\)[/tex] (considering close approximation)
So the final equivalent expressions are:
[tex]\[
252^{\frac{1}{2}}, \quad 6 \sqrt{7}
\][/tex]
