Final answer:
Expressions are compared by calculation, using rules of multiplication and exponents. Real numbers are contrasted with square roots of negative numbers being undefined in the set of real numbers, and sign patterns are considered when squaring or cubing numbers.
Step-by-step explanation:
Let's compare the expressions by evaluating each one and using the correct mathematical symbols (>, <, or =) to show their relationships.
- − (3 √ 2)^3 versus (3 √ -2)^3: The cube of a negative number is negative, so − (3 √ 2)^3 is negative, whereas (3 √ -2)^3 is not a real number because the square root of a negative number is not defined in the set of real numbers. Therefore, we cannot compare these two expressions in the set of real numbers.
- (− √ 2)^2 versus − (√ 2)^2: Both expressions evaluate to 2, so they are equal: (− √ 2)^2 = − (√ 2)^2 = 2.
- 2^3 versus − 2^3: 2^3 is positive and equals 8, while − 2^3 is negative and equals -8. Thus, 2^3 > − 2^3.
- − 2^2 versus (− 2)^2: − 2^2 is negative and equals -4, while (− 2)^2 is positive and equals 4. Therefore, − 2^2 < (− 2)^2.
- 3 √ 64 versus √ 16: 3 √ 64 is equal to 3 × 8 which is 24, and √ 16 equals 4. Hence, 3 √ 64 > √ 16.
Rules such as the multiplication of numbers with different signs and the properties of exponents are applied to determine the evaluations.