Answer :
Sure! Let's find out which of the given options are square roots of [tex]\( 576 \)[/tex]. We'll evaluate each option one by one:
### Step-by-Step Solution:
1. Option A: [tex]\( 576^{1/2} \)[/tex]
- By definition, taking the number [tex]\( 576 \)[/tex] to the power of [tex]\( 1/2 \)[/tex] means finding its square root.
- Therefore, [tex]\( 576^{1/2} \)[/tex] is indeed a square root of [tex]\( 576 \)[/tex].
2. Option B: 24
- To determine if 24 is a square root of 576, we can square 24:
[tex]\[
24^2 = 576
\][/tex]
- Therefore, 24 is indeed a square root of 576.
3. Option C: -24
- Similarly, let's check if -24 is a square root of 576. Squaring -24 gives:
[tex]\[
(-24)^2 = 576
\][/tex]
- Therefore, -24 is indeed a square root of 576.
4. Option D: 48
- Checking if 48 is a square root involves squaring 48:
[tex]\[
48^2 = 2304
\][/tex]
- Since 2304 is not equal to 576, 48 is not a square root of 576.
5. Option E: 12
- Squaring 12 gives:
[tex]\[
12^2 = 144
\][/tex]
- Since 144 is not equal to 576, 12 is not a square root of 576.
6. Option F: [tex]\(-576^{1/2}\)[/tex]
- [tex]\(-576^{1/2}\)[/tex] means finding the negative square root of 576.
- We've already established that [tex]\( 576^{1/2} = 24 \)[/tex], so [tex]\(-576^{1/2}\)[/tex] is:
[tex]\[
-576^{1/2} = -24
\][/tex]
- Thus, [tex]\(-576^{1/2}\)[/tex] is indeed a square root of 576.
### Conclusion:
The options that are square roots of 576 are:
- A: [tex]\( 576^{1/2} \)[/tex]
- B: 24
- C: -24
- F: [tex]\(-576^{1/2}\)[/tex]
Hence, the correct answers are A, B, C, and F.
### Step-by-Step Solution:
1. Option A: [tex]\( 576^{1/2} \)[/tex]
- By definition, taking the number [tex]\( 576 \)[/tex] to the power of [tex]\( 1/2 \)[/tex] means finding its square root.
- Therefore, [tex]\( 576^{1/2} \)[/tex] is indeed a square root of [tex]\( 576 \)[/tex].
2. Option B: 24
- To determine if 24 is a square root of 576, we can square 24:
[tex]\[
24^2 = 576
\][/tex]
- Therefore, 24 is indeed a square root of 576.
3. Option C: -24
- Similarly, let's check if -24 is a square root of 576. Squaring -24 gives:
[tex]\[
(-24)^2 = 576
\][/tex]
- Therefore, -24 is indeed a square root of 576.
4. Option D: 48
- Checking if 48 is a square root involves squaring 48:
[tex]\[
48^2 = 2304
\][/tex]
- Since 2304 is not equal to 576, 48 is not a square root of 576.
5. Option E: 12
- Squaring 12 gives:
[tex]\[
12^2 = 144
\][/tex]
- Since 144 is not equal to 576, 12 is not a square root of 576.
6. Option F: [tex]\(-576^{1/2}\)[/tex]
- [tex]\(-576^{1/2}\)[/tex] means finding the negative square root of 576.
- We've already established that [tex]\( 576^{1/2} = 24 \)[/tex], so [tex]\(-576^{1/2}\)[/tex] is:
[tex]\[
-576^{1/2} = -24
\][/tex]
- Thus, [tex]\(-576^{1/2}\)[/tex] is indeed a square root of 576.
### Conclusion:
The options that are square roots of 576 are:
- A: [tex]\( 576^{1/2} \)[/tex]
- B: 24
- C: -24
- F: [tex]\(-576^{1/2}\)[/tex]
Hence, the correct answers are A, B, C, and F.
