Solve 7^-4 Minus 7^-8: Negative Exponent Subtraction Problem

In this problem, we are asked to decide whether the given relation is an equality or an inequality. If it is an inequality, we must also determine its direction.

Let’s begin solving. It is clear that the two expressions are not equal, so we are dealing with an inequality. The question, then, is: which side is greater—the left-hand expression or the right-hand one?

To answer this, let’s recall the rules for evaluating inequalities with exponents. These rules state that the direction of the inequality between expressions with the same base depends on both the value of the base and the exponents, as follows:

For a base greater than one, the direction of inequality between the exponential expressions - will maintain the direction of inequality between the exponents, meaning for base: $x$, such that:

$$$x>1$(the base is always defined to be a positive number)

and exponents $a,\hspace{4pt}b$such that: $a>b$it follows that:

$x^a>x^b$

And for a base less than 1 and greater than 0, the direction of inequality between the exponential expressions - will be opposite to the direction of inequality between the exponents, meaning for base: $x$, such that:

$$$1 >x>0$(the base is always defined to be a positive number)

and exponents $a,\hspace{4pt}b$such that: $a>b$it follows that:

$x^b >x^a$

Let's return then to the problem:

We are required to determine the direction of inequality between the expressions:

$7^{-4}\text{ }_{——\text{ }}7^{-8}$The base in both expressions equals 7, meaning it's greater than one, and therefore the direction of inequality between the expressions will maintain the direction of inequality that exists between the exponents, therefore, we'll examine the exponents of the expressions in question and since it's clear that:

$-4>-8$then it follows that:

$7^{-4}\text{ }>{\text{ }}7^{-8}$

Therefore the correct answer is answer B.