Negative Exponents

Before we dive into the world of negative exponents, here's a quick recap of what exponents are.

Exponents

Exponents (also known as indices or powers) are shorthand for repeated multiplication. The exponent of a number tells us how many copies of it are multiplied together.

For example —

Exponents are shorthand for repeated multiplication

Now everything is pretty straightforward when the exponent is positive. But what if it's negative?

What Do Negative Exponents Mean?

Negative exponents are the opposite of positive exponents and imply repeated division (inverse or opposite of repeated multiplication). So 2-3 means we divide by 2, three times.

In other words, $\hspace{0.2em} 2^{-3} \hspace{0.2em}$ is the same as $\hspace{0.2em} \frac{1}{2^3} \hspace{0.2em}$ . Here's how.

\begin{align*} 2^{-3} &= 1 \div 2 \div 2 \div 2 \\[1.3em] &= \frac{1}{2 \times 2 \times 2} \\[1.3em] &= \frac{1}{2^{-3}} \end{align*}

Intuition for Negative Exponents

One way to visualize why negative exponents represent repeated division is to observe the pattern as an exponent goes up or down in steps of one.

The pattern goes on indefinitely on either side

As you can see, increase the exponent by one has the same effect as multiplying by $\hspace{0.2em} 2 \hspace{0.2em}$ (or whatever the base is). Similarly decreasing the exponent by one is the same as dividing by the base ( $\hspace{0.2em} 2 \hspace{0.2em}$ in this case).

See what happens as the exponent decreases to zero and then turns negative?

Alright, a couple more examples of how negative exponents are linked to their positive counterparts.

Negative to Positive Exponents
(Examples)

The basic idea is this.

By taking the reciprocal of the base (interchanging the numerator and the denominator) you can change the sign of its exponent – from positive to negative and the other way round.

In general,

a^{-n} = \frac{1}{a^n}

If you have any doubts at this point, don't worry. The following examples should help greatly.

Example

Simplify -

(a) \hspace{0.3cm} \frac{5}{3^{-2}}

(b) \hspace{0.3cm} \left ( \frac{4}{3} \right )^{-2}

Solution (a)

What do we do here? We have a negative exponent on 3 which is in the denominators.

To turn the exponent into positive, we will shift the base (3) to the top.

\frac{5}{ {\color{Red} 3^{-2}} } = \frac{5 \cdot {\color{Red} 3^2} }{1}

Note - We only take the reciprocal of the base of the exponent - in this case, 3. So we don't disturb 5.

Now, as 3 moves to the top, there's nothing left at the bottom - except 1, which we need not show. So,

\frac{5 \cdot {\color{Red} 3^2} }{1} = 5 \cdot {\color{Red} 9} = 45

That's it.

Solution (b)

This time, the negative exponent is applied to the fraction (all of it). So we take the reciprocal of the fraction.

\begin{align*} \left ( \frac{4}{3} \right )^{-2} &= \left ( \frac{3}{4} \right )^2 \\[1.3em] &= \frac{9}{16} \end{align*}

Simple enough.

Example

Rewrite using positive exponents only.

(a) \hspace{0.3cm} \frac{x^{-2}}{y^3}

(b) \hspace{0.3cm} \frac{x^2}{y^{-3}}

Solution (a)

Alright, it's getting more interesting.

Here, we have two variables. But y's exponent is already positive. So we don't do anything to y.

\frac{ {\color{Red} x^{-2}} }{y^3} = \frac{1}{ {\color{Red} x^{2}} y^3}

Note - When we move x to the bottom, there is nothing left at the top except 1. And when 1 is the only thing at the top, we must show it - unlike when it is at the bottom.

Solution (b)

Very similar to the last example.

\frac{x^2}{ {\color{Red} y^{-3}} } = x^2 {\color{Red} y^3}

Again, as y goes to the top, there's nothing left at the bottom (except 1, which need not be shown).

Example

Rewrite using positive exponents only.

(a) \hspace{0.3cm} -2p^{-5}{q^2}

(b) \hspace{0.3cm} \frac{-5x^{-2}}{y^{-1}}

Solution (a)

Through this example, I wanted to highlight that we are only concerned with the negative exponents. Any other negative signs remain unaffected by our moves here (for example, the frontmost negative sign in this example).

-2 {\color{Red} p^{-5}} q^2 = \frac{-2q^2}{ {\color{Red} p^5} }

Solution (b)

Same story.

\frac{-5 {\color{Red} x^{-2}} }{ {\color{Teal} y^{-1}} } = \frac{-5 {\color{Teal} y} }{ {\color{Red} x^2} }

These Two Are Not the Same

It’s important to understand that $\hspace{0.2em} {-a}^n \hspace{0.2em}$ is not the same as $\hspace{0.2em} a^{-n} \hspace{0.2em}$ .

For example, $\hspace{0.2em} {-2}^3 \hspace{0.2em}$ is not the same as $\hspace{0.2em} 2^{-3} \hspace{0.2em}$ . If we simplify them, they will give very different results.

\begin{align*} {-2}^3 &= -(2^3) \\[1.3em] &= {-8} \end{align*}

\begin{align*} 2^{-3} &= \frac{1}{2^3} \\[1.3em] &= \frac{1}{8} \end{align*}

So don't get confused between the two.

Negative Exponents – A Different Approach

In an earlier section, we tried to gain some intuition for negative exponents by analyzing how they fit the larger pattern. But there's also another simple way to think about it.

We know from the rules of exponents that -

a^{m - n} = \frac{a^m}{a^n}

Now, see what happens when we place $\hspace{0.2em} 0 \hspace{0.2em}$ in place of $\hspace{0.2em} m \hspace{0.2em}$ .

\begin{align*} a^{0 - n} &= \frac{ {\color{Red} a^0} }{a^n} \\[1.3em] \Rightarrow \hspace{0.25em} a^{-n} &= \frac{ {\color{Red} 1} }{a^n} \end{align*}

As expected, we get the same relationship between negative exponents and their positive counterparts.

And with that we come to the end of this tutorial on negative exponents. Until next time.

Footnotes

If you take a fraction and turn it upside down (interchange its numerator and denominator), you get its reciprocal. For example, the reciprocal of $\hspace{0.2em} \frac{2}{3} \hspace{0.2em}$ is $\hspace{0.2em} \frac{3}{2} \hspace{0.2em}$ , and vice-versa.

But you might ask, how can whole numbers have reciprocals when they don't have bottom numbers?

Well, every whole number can be written as a fraction with its denominator being $\hspace{0.2em} 1 \hspace{0.2em}$ . For example, $\hspace{0.2em} 3 \hspace{0.2em}$ can be written as $\hspace{0.2em} \frac{3}{1} \hspace{0.2em}$ . And so the reciprocal of $\hspace{0.2em} 3 \hspace{0.2em}$ is $\hspace{0.2em} \frac{1}{3} \hspace{0.2em}$ .