Recognize that the expression involves a negative exponent and a fractional exponent.
Recall that a negative exponent indicates the reciprocal of the base raised to the positive of that exponent. So, .
Understand that the fractional exponent represents the cube root. Therefore, is the cube root of 125.
Identify that 125 is a perfect cube, specifically .
Conclude that , so .
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Exponents and Radicals
Exponents represent repeated multiplication of a base number. A negative exponent indicates the reciprocal of the base raised to the absolute value of the exponent. For example, a base of 125 with an exponent of -1/3 means we take the reciprocal of 125 raised to the power of 1/3, which involves both negative and fractional exponent rules.
The cube root of a number is a value that, when multiplied by itself three times, gives the original number. In this case, 125 is a perfect cube, as 5 x 5 x 5 equals 125. Therefore, the cube root of 125 is 5, which is essential for evaluating the expression 125^(-1/3).
The reciprocal of a number is 1 divided by that number. When evaluating expressions with negative exponents, finding the reciprocal is crucial. For instance, the expression 125^(-1/3) translates to 1 divided by the cube root of 125, which simplifies the evaluation process and leads to the final result.