Answer :
To compare [tex]\(\sqrt{122}\)[/tex] and [tex]\(\sqrt{123}\)[/tex], let's analyze each of the given statements:
Statement A:
[tex]\[
(\sqrt{122})^2 = 122 \text{ and } (\sqrt{123})^2 = 123. \text{ Since } 122 > 123, \sqrt{122} < \sqrt{123}.
\][/tex]
This statement incorrectly states that [tex]\(122 > 123\)[/tex]. We know that [tex]\(122 < 123\)[/tex], so this statement is incorrect.
Statement B:
[tex]\[
(\sqrt{122})^2 = 122 \text{ and } (\sqrt{123})^2 = 123. \text{ Since } 122 < 123, \sqrt{122} < \sqrt{123}.
\][/tex]
This statement correctly identifies that [tex]\(122 < 123\)[/tex]. When comparing square roots, if a number is smaller, its square root is also smaller. Thus, since [tex]\(122 < 123\)[/tex], it follows that [tex]\(\sqrt{122} < \sqrt{123}\)[/tex]. Therefore, this statement is correct.
Statement C:
[tex]\[
(\sqrt{122})^2 = 122 \text{ and } (\sqrt{123})^2 = 123. \text{ Since } 122 < 123, \sqrt{122} > \sqrt{123}.
\][/tex]
This statement acknowledges that [tex]\(122 < 123\)[/tex] but incorrectly states that [tex]\(\sqrt{122}\)[/tex] is greater than [tex]\(\sqrt{123}\)[/tex]. Since [tex]\(122 < 123\)[/tex], it should be [tex]\(\sqrt{122} < \sqrt{123}\)[/tex]. Hence, this statement is incorrect.
Statement D:
[tex]\[
(\sqrt{122})^2 = 122 \text{ and } (\sqrt{123})^2 = 123. \text{ Since } 122 > 123, \sqrt{122} > \sqrt{123}.
\][/tex]
This statement repeats the error of stating [tex]\(122 > 123\)[/tex], which is false. Hence, this statement is incorrect.
After considering and evaluating each statement, the correct statement is B:
[tex]\[
(\sqrt{122})^2 = 122 \text{ and } (\sqrt{123})^2 = 123. \text{ Since } 122 < 123, \sqrt{122} < \sqrt{123}.
\][/tex]
Statement A:
[tex]\[
(\sqrt{122})^2 = 122 \text{ and } (\sqrt{123})^2 = 123. \text{ Since } 122 > 123, \sqrt{122} < \sqrt{123}.
\][/tex]
This statement incorrectly states that [tex]\(122 > 123\)[/tex]. We know that [tex]\(122 < 123\)[/tex], so this statement is incorrect.
Statement B:
[tex]\[
(\sqrt{122})^2 = 122 \text{ and } (\sqrt{123})^2 = 123. \text{ Since } 122 < 123, \sqrt{122} < \sqrt{123}.
\][/tex]
This statement correctly identifies that [tex]\(122 < 123\)[/tex]. When comparing square roots, if a number is smaller, its square root is also smaller. Thus, since [tex]\(122 < 123\)[/tex], it follows that [tex]\(\sqrt{122} < \sqrt{123}\)[/tex]. Therefore, this statement is correct.
Statement C:
[tex]\[
(\sqrt{122})^2 = 122 \text{ and } (\sqrt{123})^2 = 123. \text{ Since } 122 < 123, \sqrt{122} > \sqrt{123}.
\][/tex]
This statement acknowledges that [tex]\(122 < 123\)[/tex] but incorrectly states that [tex]\(\sqrt{122}\)[/tex] is greater than [tex]\(\sqrt{123}\)[/tex]. Since [tex]\(122 < 123\)[/tex], it should be [tex]\(\sqrt{122} < \sqrt{123}\)[/tex]. Hence, this statement is incorrect.
Statement D:
[tex]\[
(\sqrt{122})^2 = 122 \text{ and } (\sqrt{123})^2 = 123. \text{ Since } 122 > 123, \sqrt{122} > \sqrt{123}.
\][/tex]
This statement repeats the error of stating [tex]\(122 > 123\)[/tex], which is false. Hence, this statement is incorrect.
After considering and evaluating each statement, the correct statement is B:
[tex]\[
(\sqrt{122})^2 = 122 \text{ and } (\sqrt{123})^2 = 123. \text{ Since } 122 < 123, \sqrt{122} < \sqrt{123}.
\][/tex]
