Let's analyze each of the given options to see which ones are equivalent to the expression \(\sqrt{252}\):
1. Option 1: \(18 \sqrt{7}\)
First, evaluate \(18 \sqrt{7}\):
[tex]\[
\sqrt{252} \approx 15.8745
\][/tex]
[tex]\[
18 \sqrt{7} \approx 18 \cdot 2.6458 \approx 47.6244
\][/tex]
Since \(47.6244\) is not equal to \(15.8745\), \(\sqrt{252} \neq 18 \sqrt{7}\).
2. Option 2: \(126^{\frac{1}{2}}\)
Evaluate \(126^{\frac{1}{2}}\):
[tex]\[
\sqrt{252} \approx 15.8745
\][/tex]
[tex]\[
126^{\frac{1}{2}} = \sqrt{126} \approx 11.2250
\][/tex]
Since \(11.2250\) is not equal to \(15.8745\), \(\sqrt{252} \neq 126^{\frac{1}{2}}\).
3. Option 3: \(6 \sqrt{7}\)
Evaluate \(6 \sqrt{7}\):
[tex]\[
\sqrt{252} \approx 15.8745
\][/tex]
[tex]\[
6 \sqrt{7} \approx 6 \cdot 2.6458 \approx 15.8748
\][/tex]
This is extremely close to \(15.8745\), within the precision we consider here equivalent. Therefore, \(\sqrt{252} \approx 6 \sqrt{7}\).
4. Option 4: \(7 \sqrt{6}\)
Evaluate \(7 \sqrt{6}\):
[tex]\[
\sqrt{252} \approx 15.8745
\][/tex]
[tex]\[
7 \sqrt{6} \approx 7 \cdot 2.4495 \approx 17.1465
\][/tex]
Since \(17.1465\) is not equal to \(15.8745\), \(\sqrt{252} \neq 7 \sqrt{6}\).
5. Option 5: \(252^{\frac{1}{2}}\)
Evaluate \(252^{\frac{1}{2}}\):
[tex]\[
252^{\frac{1}{2}} = \sqrt{252}
\][/tex]
This matches directly with the given expression.
Thus, the expressions equivalent to \(\sqrt{252}\) are:
- \(6 \sqrt{7}\) (Option 3)
- \(252^{\frac{1}{2}}\) (Option 5)
So, the correct answers are Option 3 and Option 5.
