To determine which number will make the equation [tex]\(\square^2 = \sqrt{64}\)[/tex] true, let's go through the steps required to solve it.
1. Simplify the equation: First, we need to understand the right-hand side of the equation.
- [tex]\(\sqrt{64}\)[/tex] is the square root of 64.
- The square root of 64 is 8 (since [tex]\(8 \times 8 = 64\)[/tex]).
- Thus, [tex]\(\square^2 = 8\)[/tex].
2. Solve for the box: Next, we want to find the number that, when squared, equals 8.
- This implies [tex]\(\square = \sqrt{8}\)[/tex]. The square root of 8 is approximately 2.828.
3. Analyze the options:
- Option A: [tex]\(\sqrt{8} = 2.828\)[/tex] (approximately).
- Option B: [tex]\(\sqrt{32}\)[/tex]. The square root of 32 is approximately 5.657.
- Option C: 16.
- Option D: 32.
4. Verify each option:
- For option A ([tex]\(\sqrt{8}\)[/tex]):
[tex]\(\square = \sqrt{8}\)[/tex], which means [tex]\((\sqrt{8})^2 = 8\)[/tex]. This matches our requirement.
- For option B ([tex]\(\sqrt{32}\)[/tex]):
[tex]\(\square = \sqrt{32}\)[/tex], which means [tex]\((\sqrt{32})^2 = 32\)[/tex]. This does not equal 8.
- For option C (16):
[tex]\(\square = 16\)[/tex], which means [tex]\(16^2 = 256\)[/tex]. This does not equal 8.
- For option D (32):
[tex]\(\square = 32\)[/tex], which means [tex]\(32^2 = 1024\)[/tex]. This does not equal 8.
After verifying all the options, we find that option A, [tex]\(\sqrt{8}\)[/tex], is the correct choice.
Therefore, the number that will make the equation [tex]\(\square^2 = \sqrt{64}\)[/tex] true is:
Answer: A. [tex]\(\sqrt{8}\)[/tex].
