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Answer :
To determine the principal square root of [tex]\(-\sqrt{196}\)[/tex], let's follow these steps:
1. Evaluate [tex]\(\sqrt{196}\)[/tex]:
- The square root [tex]\(\sqrt{196}\)[/tex] is asking for a number that, when multiplied by itself, results in 196.
- We recognize that [tex]\(14 \times 14 = 196\)[/tex].
- Therefore, [tex]\(\sqrt{196} = 14\)[/tex].
2. Consider [tex]\(-\sqrt{196}\)[/tex]:
- This expression represents [tex]\(-14\)[/tex], since we previously found [tex]\(\sqrt{196} = 14\)[/tex].
3. Principal Square Root and Negative Numbers:
- The principal square root function, [tex]\(\sqrt{ }\)[/tex], by definition, only provides the non-negative root, meaning it doesn't handle negative results.
- When the root itself becomes negative, as in the expression [tex]\(-\sqrt{196}\)[/tex], there are implications for the principal square root in the realm of real numbers.
4. Dealing with [tex]\(-\sqrt{196}\)[/tex]:
- [tex]\(-\sqrt{196}\)[/tex] evaluates to [tex]\(-14\)[/tex], yet we're looking for the square root of this negative number.
- The concept of a square root producing a negative number isn't valid in real numbers because any number squared (positive or negative) results in a positive number.
5. Conclusion:
- Since forming a negative number from the square root of another number yields no possible solutions in the real number set, we conclude that [tex]\(-\sqrt{196}\)[/tex] results in no real roots.
So, when you try to find the principal square root of [tex]\(-\sqrt{196}\)[/tex], you determine that there are no possible real roots. The result is "no real roots.”
1. Evaluate [tex]\(\sqrt{196}\)[/tex]:
- The square root [tex]\(\sqrt{196}\)[/tex] is asking for a number that, when multiplied by itself, results in 196.
- We recognize that [tex]\(14 \times 14 = 196\)[/tex].
- Therefore, [tex]\(\sqrt{196} = 14\)[/tex].
2. Consider [tex]\(-\sqrt{196}\)[/tex]:
- This expression represents [tex]\(-14\)[/tex], since we previously found [tex]\(\sqrt{196} = 14\)[/tex].
3. Principal Square Root and Negative Numbers:
- The principal square root function, [tex]\(\sqrt{ }\)[/tex], by definition, only provides the non-negative root, meaning it doesn't handle negative results.
- When the root itself becomes negative, as in the expression [tex]\(-\sqrt{196}\)[/tex], there are implications for the principal square root in the realm of real numbers.
4. Dealing with [tex]\(-\sqrt{196}\)[/tex]:
- [tex]\(-\sqrt{196}\)[/tex] evaluates to [tex]\(-14\)[/tex], yet we're looking for the square root of this negative number.
- The concept of a square root producing a negative number isn't valid in real numbers because any number squared (positive or negative) results in a positive number.
5. Conclusion:
- Since forming a negative number from the square root of another number yields no possible solutions in the real number set, we conclude that [tex]\(-\sqrt{196}\)[/tex] results in no real roots.
So, when you try to find the principal square root of [tex]\(-\sqrt{196}\)[/tex], you determine that there are no possible real roots. The result is "no real roots.”
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