In order to determine what happens to the quantity of

x^{- y}

as

y

increases, let's substitute different increasing values of

y

into the expression and analyze it. We are given that

y > 0 .

Thus, we can start with substituting

1

for

y .

x^{- y} \Rightarrow x^{- 1}

Let's recall the Negative Exponent Rule, which states that negative exponents in the numerator get moved to the denominator and become positive exponents. Thus, our expression can be transformed the following way.

x^{- 1} = \frac{1}{x ^{1}} = \frac{1}{x}

Now, let's substitute

y

with another number, for example,

2 .

x^{- 2} = \frac{1}{x ^{2}}

We need to compare the two fractions that we got. Since

x

is greater than

1,

then

x^{2}

is greater than

x .

The fractions have the same numerator,

1 .

Hence, the greater the denominator is, the smaller the fraction is. We conclude that

\frac{1}{x ^{2}}

is a smaller fraction than

\frac{1}{x} .

\frac{1}{x ^{2}} < \frac{1}{x}

Now, we can substitute

y

with even greater number,

5 .

x^{- 5} = \frac{1}{x ^{5}}

x^{5}

is greater than both

x^{2}

and

x .

Therefore, the fraction

\frac{1}{x ^{5}}

is smaller than the other two fractions.

\frac{1}{x ^{5}} < \frac{1}{x ^{2}} < \frac{1}{x}

We got that the greater the value of

y

is, the smaller the fraction is. Therefore, the quantity of

x^{- y}

decreases as

y

increases.

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