Sure, let's determine which of the given options are the square roots of 81.
### Step-by-Step Solution:
1. Understanding Square Roots:
- A square root of a number [tex]\( x \)[/tex] is a number [tex]\( y \)[/tex] such that [tex]\( y^2 = x \)[/tex].
- For 81, we need to find [tex]\( y \)[/tex] such that [tex]\( y^2 = 81 \)[/tex].
2. Calculating the Positive Square Root of 81:
- [tex]\( \sqrt{81} = 9 \)[/tex]
- Because [tex]\( 9^2 = 81 \)[/tex]
3. Finding the Negative Square Root:
- Square roots can be both positive and negative.
- The negative counterpart of the positive square root is:
- [tex]\( -\sqrt{81} = -9 \)[/tex]
- Because [tex]\( (-9)^2 = 81 \)[/tex]
4. Calculating the Absolute Value of the Positive Square Root:
- The absolute value of [tex]\( 9 \)[/tex] is [tex]\( |9| \)[/tex].
- [tex]\( |9| = 9 \)[/tex]
5. Eliminating Incorrect Options:
- Option B: [tex]\( -3 \)[/tex]. Calculating [tex]\( (-3)^2 = 9 \)[/tex], not 81. So, -3 is not a square root of 81.
- Option C: [tex]\( 3 \)[/tex]. Calculating [tex]\( 3^2 = 9 \)[/tex], not 81. So, 3 is not a square root of 81.
- Option F: [tex]\( 9.5 \)[/tex]. Calculating [tex]\( 9.5^2 = 90.25 \)[/tex], not 81. So, 9.5 is not a square root of 81.
### Conclusion:
Based on our calculations, the valid square roots of 81 are:
- Option A: [tex]\( 9 \)[/tex]
- Option D: [tex]\( -9 \)[/tex]
- Option E: [tex]\( |9| = 9 \)[/tex]
So, the correct answers are:
- [tex]\[ \boxed{A, D, \text{and } E} \][/tex]
