Negative Exponents: Rules for Multiplying & Dividing

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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 = 1/(8 • 8) = 1/16 and 10-3 = 1/(10 • 10 • 10) = 1/1,000 = 0.001. It's possible to express a generalized negative exponent definition by writing: x-n = 1/xn.

TL;DR (Too Long; Didn't Read)

To multiply by a negative exponent, subtract that exponent. To divide by a negative exponent, add that exponent.

Multiplying Negative Exponents

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, x5 • x3 = x(5 +3) = x8. To see why this is true, note that x5 means (x • x • x • x •x) and x3 means (x • x • x). When you multiply these terms, you get (x • x • x • x •x • x • x • x) = x8.

A negative exponent means to divide the base raised to that power into 1. So x5 • x-3 actually means x5 • 1/x3 or (x • x • x • x • x) • 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,

xn • x-m = x(n - m)

Dividing Negative Exponents

According to the definition of a negative exponent, x-n = 1/xn. 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 1/x-n = 1/(1/xn) = xn. For example, the number x5/x-3 is equivalent to x5 • x3. You add the exponents to get x8. The rule is:

xn/x-m = x(n + m)

Examples

1. Simplify x5y4 • x-2y2

Collecting the exponents:

x(5 - 2)y(4 +2)

x3y6

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

2. Simplify (x3y-5)/(x2 y-3)

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

[(x3y-5) • y3]/ x2

x(3 - 2)y(-5 + 3)

xy-2

x/y2

3. Simplify x0y2/xy-3

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

x-1y(2 + 3)

y5/x.

References

About the Author

Chris Deziel holds a Bachelor's degree in physics and a Master's degree in Humanities, He has taught science, math and English at the university level, both in his native Canada and in Japan. He began writing online in 2010 with the goal of exploring scientific, cultural and practical topics, and at last count had reached over a hundred million readers through various sites.

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