High School

Select all items equivalent to [tex]\sqrt{24}[/tex].

A. The area of a square with side length 24 units
B. The side length of a square with area 24 square units
C. The positive number [tex]x[/tex], where [tex]x \cdot x = 24[/tex]
D. The positive number [tex]y[/tex], where [tex]y = 24 \cdot 24[/tex]
E. The edge length of a cube with volume 24 cubic units
F. The volume of a cube with edge length 24 units

Answer :

Sure, let me guide you through the question and identify which items are equivalent to [tex]\(\sqrt{24}\)[/tex].

1. Understanding [tex]\(\sqrt{24}\)[/tex]:
- [tex]\(\sqrt{24}\)[/tex] means finding a number that, when multiplied by itself, gives 24. It represents the positive number [tex]\(x\)[/tex] for which [tex]\(x \cdot x = 24\)[/tex].

Now, let's evaluate each option to see which ones match this concept:

a. The area of a square with side length 24 units:
- If a square has a side length of 24 units, its area is [tex]\(24 \cdot 24 = 576\)[/tex].
- This is not equivalent to [tex]\(\sqrt{24}\)[/tex].

b. The side length of a square with area 24 square units:
- If the area of a square is 24, then the side length is [tex]\(\sqrt{24}\)[/tex].
- This matches the definition of [tex]\(\sqrt{24}\)[/tex].

c. The positive number [tex]\(x\)[/tex], where [tex]\(x \cdot x = 24\)[/tex]:
- This is a direct definition of [tex]\(\sqrt{24}\)[/tex], as [tex]\(x\)[/tex] would be [tex]\(\sqrt{24}\)[/tex].

d. The positive number [tex]\(y\)[/tex], where [tex]\(y = 24 \cdot 24\)[/tex]:
- Here, [tex]\(y\)[/tex] equals 576, which is not [tex]\(\sqrt{24}\)[/tex].

e. The edge length of a cube with volume 24 cubic units:
- The edge length of a cube with volume 24 is [tex]\(\sqrt[3]{24}\)[/tex] (the cube root of 24), not [tex]\(\sqrt{24}\)[/tex].

f. The volume of a cube with edge length 24 units:
- If the edge length is 24, then the volume is [tex]\(24 \cdot 24 \cdot 24 = 24^3\)[/tex], which is much larger than [tex]\(\sqrt{24}\)[/tex].

So, the items equivalent to [tex]\(\sqrt{24}\)[/tex] are:

- b. The side length of a square with area 24 square units
- c. The positive number [tex]\(x\)[/tex], where [tex]\(x \cdot x = 24\)[/tex]

These are the options equivalent to [tex]\(\sqrt{24}\)[/tex].