Negative Exponents: Rules For Multiplying & Dividing

If you've been doing math for a while, you have probably come across exponents. An exponent is a number, which is called the base, followed by another number usually written in superscript. The second number is the exponent or the power. It tells you how many time to multiply the base by itself. For example, 8² means to multiply 8 by itself twice to get 16, and 10³ means 10 × 10 × 10 = 1,000. When you have negative exponents, the negative exponent rule dictates that, instead of multiplying the base the indicated number of times, you divide the base into 1 that number of times. So

\(8^{ -2} = \frac{1}{8 × 8} = \frac{1}{64} \text{ and } 10^{-3} = \frac{1}{10 × 10 × 10} = \frac{1}{1,000} = 0.001\)

It's possible to express a generalized negative exponent definition by writing:

\(x^{-n} = \frac{1}{x^n}\)

Keeping in mind that you can multiply exponents only if they have the same base, the general rule for multiplying two numbers raised to exponents is to add the exponents. For example:

\(x^5 × x^3 = x^{(5 +3)} = x^8\)

To see why this is true, note that ​x​⁵ means (​x​ × ​x​ × ​x​ × ​x​ × ​x​) and ​x​³ means (​x​ × ​x​ × ​x​). When you multiply these terms, you get (​x​ × ​x​ × ​x​ × ​x​ × ​x​ × ​x​ × ​x​ × ​x​) = ​x​⁸.

A negative exponent means to divide the base raised to that power into 1. So

\(x^5 × x^{ -3} = x^5 × \frac{1}{x^3} = (x × x × x × x × x) × \frac{1}{x × x × x}\)

This is a simple division. You can cancel three of the x's, leaving (x × x) or x². In other words, you when you multiply by a negative exponent, you still add the exponent, but since it's negative, this is equivalent to subtracting it. In general,

\(x^n × x^{-m} = x^{(n – m)}\)

According to the definition of a negative exponent:

\(x^{-n} = \frac{1}{x^n}\)

When you divide by a negative exponent, it's equivalent to multiplying by the same exponent, only positive. To see why this is true, consider

\(\frac{1}{x^{-n}} = \frac{1}{1/x^n} = x^n\)

For example, the number

\(\frac{x^5}{x^{-3}} = x^5 × x^3\)

You add the exponents to get ​x​⁸. The rule is:

\(\frac{x^n}{x^{-m}} = x^{(n + m)}\)

1\. Simplify

\(x^5y^4 × x^{-2}y^2\)

Collecting the exponents:

\(x^{(5 – 2)}y^{(4 +2)} = x^3y^6\)

You can only manipulate exponents if they have the same base, so you can't simplify any further.

2\. Simplify

\(\frac{x^3y^{-5}}{x^2 y^{-3 }}\)

Dividing by a negative exponent is equivalent to multiplying by the same positive exponent, so you can rewrite this expression:

\(\begin{aligned}
\frac{(x^3y^{-5}) × y^3}{ x^2} &= x^{(3 – 2)}y^{(-5 + 3)} \
&= xy^{-2} \
&=\frac{x}{y^2}
\end{aligned}\)

3\. Simplify

\(\frac{x^0y^2}{xy^{-3}}\)

Any number raised to an exponent of 0 is 1, so you can rewrite this expression to read:

\(x^{-1}y^{(2 + 3)} =\frac{y^5}{x}\)