Answer :
To determine which statement is justified by the equation [tex]\(14^2 = 196\)[/tex], let's evaluate each statement separately:
1. Statement: "14 is a perfect square."
- A perfect square is a number that can be expressed as the square of an integer. Since [tex]\(14\)[/tex] is not an integer squared (for instance, [tex]\(3^2 = 9\)[/tex] and [tex]\(4^2 = 16\)[/tex]), [tex]\(14\)[/tex] itself is not a perfect square. Therefore, this statement is not justified.
2. Statement: "196 is a perfect square."
- A perfect square is a number that can be obtained by squaring an integer. For example, [tex]\(14^2 = 196\)[/tex], which means [tex]\(196\)[/tex] is indeed [tex]\(14\)[/tex] squared, and since [tex]\(14\)[/tex] is an integer, [tex]\(196\)[/tex] is a perfect square. Thus, this statement is justified.
3. Statement: "[tex]\(\sqrt{14} = 196\)[/tex]"
- The square root of [tex]\(14\)[/tex] is a number that, when squared, equals [tex]\(14\)[/tex]. The value of [tex]\(\sqrt{14}\)[/tex] is approximately [tex]\(3.74\)[/tex], which is not equal to [tex]\(196\)[/tex]. Therefore, this statement is incorrect.
Based on these evaluations, the correct statement justified by [tex]\(14^2 = 196\)[/tex] is:
- "196 is a perfect square."
1. Statement: "14 is a perfect square."
- A perfect square is a number that can be expressed as the square of an integer. Since [tex]\(14\)[/tex] is not an integer squared (for instance, [tex]\(3^2 = 9\)[/tex] and [tex]\(4^2 = 16\)[/tex]), [tex]\(14\)[/tex] itself is not a perfect square. Therefore, this statement is not justified.
2. Statement: "196 is a perfect square."
- A perfect square is a number that can be obtained by squaring an integer. For example, [tex]\(14^2 = 196\)[/tex], which means [tex]\(196\)[/tex] is indeed [tex]\(14\)[/tex] squared, and since [tex]\(14\)[/tex] is an integer, [tex]\(196\)[/tex] is a perfect square. Thus, this statement is justified.
3. Statement: "[tex]\(\sqrt{14} = 196\)[/tex]"
- The square root of [tex]\(14\)[/tex] is a number that, when squared, equals [tex]\(14\)[/tex]. The value of [tex]\(\sqrt{14}\)[/tex] is approximately [tex]\(3.74\)[/tex], which is not equal to [tex]\(196\)[/tex]. Therefore, this statement is incorrect.
Based on these evaluations, the correct statement justified by [tex]\(14^2 = 196\)[/tex] is:
- "196 is a perfect square."
