To determine if the given numbers are perfect cubes, we need to verify whether there exists an integer whose cube equals the given number. Let's evaluate each number step-by-step:
### a. 675
- 675 is not a perfect cube.
- The nearest cube root rounded is not an integer that satisfies the condition [tex]\( x^3 = 675 \)[/tex].
### b. 1728
- 1728 is a perfect cube.
- 12 cubed is [tex]\( 12^3 = 12 \times 12 \times 12 = 1728 \)[/tex], so 1728 is indeed a perfect cube.
### c. 157464
- 157464 is a perfect cube.
- 54 cubed is [tex]\( 54^3 = 54 \times 54 \times 54 = 157464 \)[/tex], confirming that 157464 is a perfect cube.
### d. 2197
- 2197 is a perfect cube.
- 13 cubed is [tex]\( 13^3 = 13 \times 13 \times 13 = 2197 \)[/tex], so 2197 is a perfect cube.
### e. 4096
- 4096 is a perfect cube.
- 16 cubed is [tex]\( 16^3 = 16 \times 16 \times 16 = 4096 \)[/tex], confirming that 4096 is a perfect cube.
### f. 6859
- 6859 is a perfect cube.
- 19 cubed is [tex]\( 19^3 = 19 \times 19 \times 19 = 6859 \)[/tex], so 6859 is a perfect cube.
### g. 526
- 526 is not a perfect cube.
- The nearest cube root rounded is not an integer that satisfies the condition [tex]\( x^3 = 526 \)[/tex].
### Summary
- 675: Not a perfect cube (False)
- 1728: Perfect cube (True)
- 157464: Perfect cube (True)
- 2197: Perfect cube (True)
- 4096: Perfect cube (True)
- 6859: Perfect cube (True)
- 526: Not a perfect cube (False)
So, from our detailed check, the results are as follows:
- 675: False
- 1728: True
- 157464: True
- 2197: True
- 4096: True
- 6859: True
- 526: False
