Fill in the boxes so that each statement is true. (More than...

Fill in the boxes so that each statement is true. (More than one answer is possible for these exercises).

Determine whether each statement is true or false.
A. $5^{-1}<5^{-2}$
B. $\left(\frac{1}{5}\right)^{-1}<\left(\frac{1}{5}\right)^{-2}$
C. $a^{-1}<a^{-2}$ for all nonzero numbers.

Key Concepts

Inequality Reversal When Inverting

When both sides of an inequality are positive, taking reciprocals will reverse the direction of the inequality. This is because the function f(x) = 1/x is a decreasing function on the set of positive numbers. This property must be carefully considered when comparing expressions that have been inverted due to negative exponents.

Domain Considerations in Exponentiation

It is essential to note that exponentiation rules, including those for negative exponents, only apply when the base is nonzero, since division by zero is undefined. In the context of comparing expressions with negative exponents, ensuring that the variable or constant involved is in the appropriate domain (typically nonzero and considering its sign) is necessary to avoid incorrect conclusions about the inequality.

Negative Exponents

A negative exponent indicates that the base should be taken as the reciprocal raised to the corresponding positive exponent. For example, a?¹ equals 1/a. This rule is fundamental in simplifying expressions involving exponents and is broadly applicable regardless of whether the base is greater than or less than one.

Reciprocal Relationships

Understanding reciprocals is crucial when dealing with negative exponents. Since a negative exponent converts a number to its reciprocal, comparing expressions with negative exponents often involves comparing their corresponding reciprocal values. This concept is particularly important in determining the order of magnitude of numbers, as taking the reciprocal of a number greater than one yields a number less than one, and vice versa.

Transcript

Please give Ace some feedback