Answer :
To determine which items are equivalent to [tex]\(\sqrt{24}\)[/tex], let's go through each option and analyze them:
A. The area of a square with side length 24 units:
- The area of a square is calculated by squaring the side length. So, if a square has a side length of 24 units, its area is [tex]\(24 \times 24 = 576\)[/tex] square units. This is not equivalent to [tex]\(\sqrt{24}\)[/tex].
B. The side length of a square with area 24 square units:
- To find the side length of a square when you know its area, you take the square root of the area. Therefore, the side length is [tex]\(\sqrt{24}\)[/tex], which matches [tex]\(\sqrt{24}\)[/tex].
C. The positive number [tex]\(x\)[/tex], where [tex]\(x \cdot x = 24\)[/tex]:
- If [tex]\(x \cdot x = 24\)[/tex], then [tex]\(x\)[/tex] is the square root of 24. Therefore, [tex]\(x\)[/tex] equals [tex]\(\sqrt{24}\)[/tex], which is [tex]\(\sqrt{24}\)[/tex].
D. The positive number [tex]\(y\)[/tex], where [tex]\(y = 24 \cdot 24\)[/tex]:
- Here, [tex]\(y\)[/tex] is calculated as [tex]\(24 \times 24 = 576\)[/tex]. This is not the same as [tex]\(\sqrt{24}\)[/tex].
E. The edge length of a cube with volume 24 cubic units:
- To find the edge length of a cube when you know its volume, you would take the cube root of the volume. The cube root of 24 is not equal to [tex]\(\sqrt{24}\)[/tex].
F. The volume of a cube with edge length 24 units:
- The volume of a cube is calculated by cubing the edge length. So, with an edge length of 24, the volume is [tex]\(24^3 = 13,824\)[/tex] cubic units. This is not equivalent to [tex]\(\sqrt{24}\)[/tex].
From this analysis, the options equivalent to [tex]\(\sqrt{24}\)[/tex] are B and C.
A. The area of a square with side length 24 units:
- The area of a square is calculated by squaring the side length. So, if a square has a side length of 24 units, its area is [tex]\(24 \times 24 = 576\)[/tex] square units. This is not equivalent to [tex]\(\sqrt{24}\)[/tex].
B. The side length of a square with area 24 square units:
- To find the side length of a square when you know its area, you take the square root of the area. Therefore, the side length is [tex]\(\sqrt{24}\)[/tex], which matches [tex]\(\sqrt{24}\)[/tex].
C. The positive number [tex]\(x\)[/tex], where [tex]\(x \cdot x = 24\)[/tex]:
- If [tex]\(x \cdot x = 24\)[/tex], then [tex]\(x\)[/tex] is the square root of 24. Therefore, [tex]\(x\)[/tex] equals [tex]\(\sqrt{24}\)[/tex], which is [tex]\(\sqrt{24}\)[/tex].
D. The positive number [tex]\(y\)[/tex], where [tex]\(y = 24 \cdot 24\)[/tex]:
- Here, [tex]\(y\)[/tex] is calculated as [tex]\(24 \times 24 = 576\)[/tex]. This is not the same as [tex]\(\sqrt{24}\)[/tex].
E. The edge length of a cube with volume 24 cubic units:
- To find the edge length of a cube when you know its volume, you would take the cube root of the volume. The cube root of 24 is not equal to [tex]\(\sqrt{24}\)[/tex].
F. The volume of a cube with edge length 24 units:
- The volume of a cube is calculated by cubing the edge length. So, with an edge length of 24, the volume is [tex]\(24^3 = 13,824\)[/tex] cubic units. This is not equivalent to [tex]\(\sqrt{24}\)[/tex].
From this analysis, the options equivalent to [tex]\(\sqrt{24}\)[/tex] are B and C.
