Final answer:
The distinction between (-5)⁴ equalling 625 and -5⁴ equalling -625 is due to the order of operations and the even exponent property; parentheses dictate that the negative base is raised to the power, resulting in a positive number for even exponents. Option A,B is correct.
Step-by-step explanation:
The difference between (-5)⁴ and -5⁴ lies primarily in the order of operations. When we have (-5)⁴, the parentheses indicate that we are raising -5 as a whole to the fourth power. According to the even exponent property, when we raise a negative number to an even power, the result is positive:
(-5) × (-5) × (-5) × (-5) = 625
However, with -5⁴, the absence of parentheses means that we only raise 5 to the fourth power and then apply the negative sign, as dictated by the order of operations:
- (5 × 5 × 5 × 5) = -625
The reason why (-5)4 is equal to 625 while -54 is equal to -625 is because of the even exponent property and the negative base property
The even exponent property states that when a negative base is raised to an even exponent, the result is always positive. In this case, (-5)4 is (-5) x (-5) x (-5) x (-5) = 625.
On the other hand, the negative base property states that when a negative base is raised to an odd exponent, the result is always negative. In this case, -54 is -5 x -5 x -5 x -5 = -625.