Complete the table by indicating whether the numerical value in each equation is a perfect square. If it isn't a perfect square, indicate which perfect squares it is between.
| Equation | Is it a perfect square? | If not, which perfect squares does it lie between? |
|---|---|---|
| [tex]x^2=16[/tex] | | |
| [tex]x^2=8[/tex] | | |
| [tex]x^2=20[/tex] | | |
| [tex]x^2=49[/tex] | | |
| [tex]x^2=121[/tex] | | |
| [tex]x^2=80[/tex] | | |
Answer (2)
The completed table is above.
Explanation
Problem Analysis We are given a table with equations and asked to determine if the numerical value in each equation is a perfect square (or cube, depending on the equation). If it is not a perfect square/cube, we need to find the perfect squares/cubes it lies between.
Equation 1: x 2 = 16 For the equation x 2 = 16 , we find that x = 16 = 4 . Since 4 is an integer, 16 is a perfect square.
Equation 2: x 2 = 8 For the equation x 2 = 8 , we find that x = 8 ≈ 2.828 . Since x is not an integer, 8 is not a perfect square. It lies between the perfect squares 2 2 = 4 and 3 2 = 9 .
Equation 3: x 3 = 20 For the equation x 3 = 20 , we find that x = 3 20 ≈ 2.714 . Since x is not an integer, 20 is not a perfect cube. It lies between the perfect cubes 2 3 = 8 and 3 3 = 27 .
Equation 4: x 2 = 49 For the equation x 2 = 49 , we find that x = 49 = 7 . Since 7 is an integer, 49 is a perfect square.
Equation 5: x 2 = 121 For the equation x 2 = 121 , we find that x = 121 = 11 . Since 11 is an integer, 121 is a perfect square.
Equation 6: x 2 = 80 For the equation x 2 = 80 , we find that x = 80 ≈ 8.944 . Since x is not an integer, 80 is not a perfect square. It lies between the perfect squares 8 2 = 64 and 9 2 = 81 .
Final Answer Therefore, the completed table is as follows:
Equation
Is it a perfect square?
If not, which perfect squares does it lie between?
x 2 = 16
Yes
x 2 = 8
No
4 and 9
x 3 = 20
No
8 and 27
x 2 = 49
Yes
x 2 = 121
Yes
x 2 = 80
No
64 and 81
Examples
Understanding perfect squares and cubes is useful in various real-life situations. For example, when designing a square garden with an area of 16 square meters, knowing that 16 is a perfect square helps determine that each side should be 4 meters long. Similarly, if you're packing items into a cubic box and want to know if you can perfectly fill a volume of 27 cubic units, recognizing that 27 is a perfect cube tells you that you can arrange the items in a 3x3x3 cube. These concepts are also crucial in more advanced fields like engineering and computer graphics, where spatial reasoning and efficient packing algorithms are essential.
The table indicates whether each equation represents a perfect square. The equations for 16, 49, and 121 are perfect squares, while 8, 20, and 80 are not, with specific perfect squares they lie between listed. The completed table accurately summarizes these findings.
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