To determine which statement correctly describes the expression [tex]\(\left|x^3\right|+5\)[/tex], we need to analyze each option:
1. Option A: the sum of the absolute value of three times a number and 5
- This would be described by the expression [tex]\(\left|3x\right| + 5\)[/tex], where [tex]\(\left|3x\right|\)[/tex] represents the absolute value of three times a number. Clearly, [tex]\(\left|x^3\right|\)[/tex] is not [tex]\(\left|3x\right|\)[/tex]. This option is incorrect.
2. Option B: the cube of the sum of a number and 5
- This implies the expression [tex]\((x + 5)^3\)[/tex], which takes a number, adds 5 to it, and then cubes the result. The form [tex]\(\left|x^3\right|+5\)[/tex] is different as it involves taking the cube of [tex]\(x\)[/tex], the absolute value of that result, and then adding 5. This option is incorrect.
3. Option C: 5 more than the absolute value of the cube of a number
- The expression [tex]\(\left|x^3\right| + 5\)[/tex] first takes the cube of [tex]\(x\)[/tex], then takes the absolute value, and finally adds 5. This perfectly matches the description "5 more than the absolute value of the cube of a number." Therefore, this option is correct.
4. Option D: the absolute value of three times a number added to 5
- This would be described by the expression [tex]\(\left|3x + 5\right|\)[/tex], where the absolute value is taken after multiplying by 3 and adding 5. Again, this is not our given expression [tex]\(\left|x^3\right| + 5\)[/tex]. This option is incorrect.
Based on the analysis of each option, Option C: 5 more than the absolute value of the cube of a number is the correct statement that describes the expression [tex]\(\left|x^3\right| + 5\)[/tex].
So, the correct answer is Option C.
