Negative Exponents in Fractions – Rule and Examples

EXAMPLE 1

Simplify to the expression $latex {{3}^{-2}}$.

Solution: We know that the negative exponent means that the base belongs to the other side of the fraction. But we see that there is no fraction line. However, we know that this can be written as a fraction with denominator 1.

Therefore, we start by converting the expression to a fraction in the way that all expressions can be converted to fractions: by putting it over the 1. Therefore, we have the following:

$latex {{3}^{{-2}}}=\frac{{{{3}^{{-2}}}}}{1}=\frac{1}{{{{3}^{2}}}}=\frac{1}{9}$

EXAMPLE 2

Write a $latex x^{-3}$ using only positive exponents.

Solution: Again, we know that we have to swap the base to the other side of the fraction and change the exponent from negative to positive. We form a fraction by putting the expression over 1.

Thus, we start by converting the expression to a fraction where the denominator is 1 and then we simplify. Therefore, we have the following:

$latex {{x}^{{-3}}}=\frac{{{{x}^{{-3}}}}}{1}=\frac{1}{{{{x}^{3}}}}$

EXAMPLE 1

• Simplify the fraction $latex \frac{{{{4}^{{-2}}}}}{{{{8}^{{-2}}}}}$.

Solution: We can apply the negative exponent rule separately to the numerator and denominator and then simplify the resulting expression. Therefore, we have:

$latex \frac{{{{4}^{{-2}}}}}{{{{8}^{{-2}}}}}=\frac{1}{{{{4}^{2}}}}\times \frac{{{{8}^{2}}}}{1}$

$latex =\frac{{{{8}^{{2}}}}}{{{{4}^{{2}}}}}$

$latex =\frac{64}{16}$

$latex =4$

We have that $latex \frac{{{{4}^{{-2}}}}}{{{{8}^{{-2}}}}}$ is equivalent to 4.

EXAMPLE 2

• Simplify the fraction $latex \frac{{3{{x}^{{-2}}}y}}{{xy}}$.

Solution: In this case, only the expression $latex {{{x}^{{-2}}}}$ has a negative exponent. Therefore, we change that expression from the numerator to the denominator and make it positive:

$latex \frac{{3{{x}^{{-2}}}y}}{{xy}}=\frac{{3y}}{{{{x}^{2}}\left( {xy} \right)}}$

$latex =\frac{{3y}}{{{{x}^{3}}y}}$

Now, we cancel the like terms in both the numerator and the denominator:

$latex \frac{{3y}}{{{{x}^{3}}y}}=\frac{{3}}{{{{x}^{3}}}}$

EXAMPLE 1

• Simplify the fraction $latex \frac{{{{4}^{{-\frac{1}{2}}}}}}{{{{6}^{{-2}}}}}$.

Solution: In the numerator we have a negative fractional exponent and in the denominator we have a negative exponent. Therefore, we start by applying the rule of negative exponents:

$latex \frac{{{{4}^{{-\frac{1}{2}}}}}}{{{{6}^{{-2}}}}}=\frac{{{{6}^{2}}}}{{{{4}^{{\frac{1}{2}}}}}}$

Now we have positive exponents in both the numerator and denominator, but we have a fractional exponent in the denominator, so we apply the fractional exponents rule there:

$latex =\frac{{{{6}^{2}}}}{{\sqrt{4}}}$

Now, we apply the exponents and simplify:

$latex =\frac{{36}}{2}$

$latex =18$

EXAMPLE 2

• Simplify the expression $$\frac{{{{x}^{{-\frac{3}{2}}}}}}{{{{{16}}^{{-\frac{1}{4}}}}{{y}^{{-\frac{1}{2}}}}}}$$

Solution: All the exponents are negative, so we start by applying the negative exponents rule:

$$\frac{{{{x}^{{-\frac{3}{2}}}}}}{{{{{16}}^{{-\frac{1}{4}}}}{{y}^{{-\frac{1}{2}}}}}}=\frac{{{{{16}}^{{\frac{1}{4}}}}{{y}^{{\frac{1}{2}}}}}}{{{{x}^{{\frac{3}{2}}}}}}$$

Now, we have positive exponents in the numerator and in the denominator. We can apply the fractional exponent rule:

$latex =\frac{{\sqrt[4]{{16}}\sqrt{y}}}{{\sqrt{{{{x}^{3}}}}}}$

Finally, we simplify:

$latex =\frac{{2~\sqrt{y}}}{{\sqrt{{{{x}^{3}}}}}}$

EXAMPLE 1

• Simplify the expression $latex \frac{{{{x}^{2}}}}{{{{x}^{{-3}}}}}$.

Solution: This expression only has the denominator with a negative exponent. Therefore, we move the denominator to the numerator with a positive exponent :

$latex \frac{{{{x}^{2}}}}{{{{x}^{{-3}}}}}=\frac{{{{x}^{2}}{{x}^{3}}}}{1}$

Now, we only have positive exponents and we can apply the product of exponents rule to simplify:

$latex ={{x}^{{2+3}}}={{x}^{5}}$

EXAMPLE 2

• Simplify the expression $$\frac{{{{{25}}^{{-\frac{1}{2}}}}}}{{{{{16}}^{{-\frac{1}{4}}}}}}$$

Solution: In this case, we have fractional negative exponents. Therefore, we start by applying the rule of negative exponents to change from the numerator to the denominator and vice versa and change from negative to positive:

$latex \frac{{{{{25}}^{{-\frac{1}{2}}}}}}{{{{{16}}^{{-\frac{1}{4}}}}}}=\frac{{{{{16}}^{{\frac{1}{4}}}}}}{{{{{25}}^{{\frac{1}{2}}}}}}$

Now, we have positive exponents in both the numerator and the denominator. Thus, we can apply the rule of fractional exponents to form radicals:

$latex =\frac{{\sqrt[4]{{16}}}}{{\sqrt{{25}}}}$

Finally, we calculate the radicals:

$latex =\frac{2}{5}$

EXAMPLE 3

• Rewrite the expression $latex {{\left( {\frac{{{{x}^{{-3}}}}}{{{{y}^{{-2}}}}}} \right)}^{{-2}}}$ only using positive exponents.

Solution: We start by noting that we have a negative exponent outside the parentheses. This means that the numerator must be moved down and the denominator must be moved up. That is, we must flip the fraction that is inside the parentheses:

$latex {{\left( {\frac{{{{x}^{{-3}}}}}{{{{y}^{{-2}}}}}} \right)}^{{-2}}}={{\left( {\frac{{{{y}^{{-2}}}}}{{{{x}^{{-3}}}}}} \right)}^{2}}$

Now that we have a positive exponent, we use the power of a power rule to apply the exponent separately to the numerator and denominator:

$latex =\frac{{{{{\left( {{{y}^{{-2}}}} \right)}}^{2}}}}{{{{{\left( {{{x}^{{-3}}}} \right)}}^{2}}}}$

$latex =\frac{{{{y}^{{-4}}}}}{{{{x}^{{-6}}}}}$

Finally, we use the negative exponents rule again and flip the fraction:

$latex =\frac{{{{x}^{6}}}}{{{{y}^{4}}}}$