The Rules of Dividing Exponents

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Exponents come up a lot in mathematics. Whether you’re simplifying algebraic equations, rearranging an equation or just completing calculations, you’re bound to encounter them eventually. The good news is that there are some simple rules for dealing with exponents, and you’ll be able to navigate problems involving them with ease once you pick them up. When dividing exponents, the basic rule for exponents with the same base is you subtract the exponent in the denominator from the one in the numerator. There’s more to learn, but this is the basic rule.

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

To divide exponents in the same base, subtract the exponent on the second base (the denominator in a fraction) from the one on the first (the numerator in a fraction).

The general rule is: xa ÷ xb = x(a b)

You can only use this rule when the base is the same. If you encounter expressions with different bases, the only way you can simplify them is by using the general rule on the parts with matching bases.

Understanding Exponents

“Exponent” is a name for the “power” that a certain number is raised to. In the term ​xb, the ​b​ is the exponent. You’ve probably encountered exponents in different situations before – perhaps in the formula for the area of a circle: ​A​ = π​r2 where the exponent is 2 or in the form of squared numbers such as 32 = 9. The latter example helps you understand what exponents mean: 3 × 3 = 32 = 9. In the same way, 33 = 3 × 3 × 3 = 27. It’s a shorthand way of saying how many times a number or symbol is multiplied by itself. Using the generic version, ​xb​, the name for ​x​ is the “base.” In 32, 3 is the base, and in ​r2, ​r​ is the base.

The Rules for Exponents: Multiplying and Dividing in the Same Base

Multiplying and dividing numbers with exponents is easy once you know two basic exponent rules. Multiplying is a bit easier to understand. If you have ​y3 × ​y2, you can write it out in full to understand what is going on:

y^3 × y^2 = (y × y × y) × (y × y) = y × y × y × y × y = y^5

In a shorter form, this is just:

y^3 × y^2 = y^5

All you do to multiply exponents is add the two numbers in the exponents and put them over the same shared base. The apparently complicated problem is just simple addition. Dividing exponents can be understood in the same way:

y^3 ÷ y^2 = \frac{y × y × y}{y × y}

Two of the ​y​s in the fraction cancel out. So this leaves ​y3 ÷ ​y2 = ​y1 = ​y​. All you wind up doing when dividing exponents is subtracting the second exponent from the first. If they’re formatted like a fraction, you subtract the exponent in the denominator from the exponent in the numerator:

\frac{y^4}{y^2 } = y^{(4-2)} = y^2

In the general form, the rule for multiplication is:

x^a × x^b = x^{(a + b)}

The rule for division is:

x^a ÷ x^b = x^{(a − b)}

Dividing Exponents in Mixed Bases

When you do algebra with exponents, in many situations there are different bases in the equation. For example, you might encounter ​x2y3÷ ​x3y2. You can only work with exponents if they have the same base, so you work with the ​x​ parts and the ​y​ parts separately:

x^2y^3÷x^3y^2 = x^{(2-3)}y^{(3-2)} = x^{-1}y^1

In reality, ​y1 is just ​y​, but it’s shown here for clarity. Note that it’s possible to have negative exponents as well as positive ones. In this case,

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

and in the same way

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

You can’t simplify the expressions more than this, so this is all you need to do.

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