To determine which number, when placed in the box, will make the equation [tex]\(\square^2 = \sqrt{64}\)[/tex] true, we need to follow these steps:
1. Calculate [tex]\(\sqrt{64}\)[/tex]:
The square root of 64 is calculated as:
[tex]\[
\sqrt{64} = 8
\][/tex]
2. Assess the given options to see which, when squared, equals 8:
Let's evaluate each option:
Option A: [tex]\(\sqrt{8}\)[/tex]
First, calculate [tex]\(\sqrt{8}\)[/tex]:
[tex]\[
\sqrt{8} = 2.8284271247461903
\][/tex]
Now, square [tex]\(\sqrt{8}\)[/tex]:
[tex]\[
(\sqrt{8})^2 = (2.8284271247461903)^2 = 8
\][/tex]
Option B: [tex]\(\sqrt{32}\)[/tex]
First, calculate [tex]\(\sqrt{32}\)[/tex]:
[tex]\[
\sqrt{32} = 5.656854249492381
\][/tex]
Now, square [tex]\(\sqrt{32}\)[/tex]:
[tex]\[
(\sqrt{32})^2 = (5.656854249492381)^2 = 32
\][/tex]
Option C: 16
Square 16:
[tex]\[
16^2 = 256
\][/tex]
Option D: 32
Square 32:
[tex]\[
32^2 = 1024
\][/tex]
3. Compare the results to [tex]\(\sqrt{64} = 8\)[/tex]:
- [tex]\((\sqrt{8})^2 = 8\)[/tex]
- [tex]\((\sqrt{32})^2 = 32\)[/tex]
- [tex]\(16^2 = 256\)[/tex]
- [tex]\(32^2 = 1024\)[/tex]
We see only one number, [tex]\(\sqrt{8} = 2.8284271247461903\)[/tex], when squared, equals 8.
Therefore, the correct answer is:
[tex]\[
\boxed{\sqrt{8}}
\][/tex]
