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Which expressions are equivalent to the given expression? [tex]\sqrt{252}[/tex]

A. [tex]18 \sqrt{7}[/tex]
B. [tex]126^{\frac{1}{2}}[/tex]
C. [tex]6 \sqrt{7}[/tex]
D. [tex]7 \sqrt{6}[/tex]
E. [tex]252^{\frac{1}{2}}[/tex]


Sagot :

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.
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