4.2 The Quotient Property, Negative Exponents, Zero Exponents, and Scientific Notation

Learning Objectives

After completing this section, you should be able to:

Simplify expressions using the Quotient Property for Exponents.
Explain how negative exponents arise when dividing powers.
Simplify expressions using zero exponents.
Apply multiple properties in one problem.
Convert between standard and scientific notation.

The Quotient Property for Exponents

In the last section, we multiplied powers and learned to add exponents.

Now we’ll see what happens when powers are divided.

Rule: Quotient of Powers

When dividing powers with the same base, subtract the exponents.

$\frac{a^m}{a^n}=a^{m-n}\quad (a\ne 0)$

Example 4.2.1 — Quotient of powers

Simplify: $\frac{x^7}{x^3}$

Solution

$\frac{x^7}{x^3}=x^{7-3}=x^4$

Example 4.2.2 — Quotient with coefficients

Simplify: $\frac{12x^6}{3x^2}$

Solution

Divide coefficients, then subtract exponents:

$\frac{12x^6}{3x^2}=\left(\frac{12}{3}\right)x^{6-2}=4x^4$

Example 4.2.3 — Quotient with two variables

Simplify: $\frac{m^4n^3}{m^2n^5}$

Solution

$\frac{m^4n^3}{m^2n^5}=m^{4-2}n^{3-5}=m^2n^{-2}$

Try It

Negative Exponents

Rule: Negative Exponents

$a^{-n}=\frac{1}{a^n}\quad (a\ne 0)$

If the base is already in the denominator, a negative exponent moves it to the numerator:

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

Let’s look closer at why negative exponents occur.

Example 4.2.5 — Why negative exponents happen

Explore: $\frac{5^2}{5^6}$

Solution

Step 1: Use the Quotient Property.

$\frac{5^2}{5^6}=5^{2-6}=5^{-4}$

Step 2: Interpret the negative exponent by expanding and canceling factors.

$\frac{5\cdot 5}{5\cdot 5\cdot 5\cdot 5\cdot 5\cdot 5}=\frac{1}{5\cdot 5\cdot 5\cdot 5}=\frac{1}{5^4}$

So $5^{-4}=\frac{1}{5^4}$ .

Generalization: If a smaller power is divided by a larger one, the result has a negative exponent because more factors remain in the denominator.

Example 4.2.6 — Rewrite with positive exponents

Simplify: $2x^{-3}$

Solution

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

Tip: Ask “What is the base?” Here the exponent applies only to $x$ , not $2x$ .

Example 4.2.7 — Multiple negative exponents

Simplify: $4a^{-2}b^{-3}$

Solution

$4a^{-2}b^{-3}=4\cdot\frac{1}{a^2}\cdot\frac{1}{b^3}=\frac{4}{a^2b^3}$

Tip: The bases are $a$ and $b$ ; the $4$ is just a coefficient.

Example 4.2.8 — Power with negative exponent inside

Simplify: $(m^{-2}n^3)^2$

Solution

$(m^{-2}n^3)^2=m^{-4}n^6=\frac{n^6}{m^4}$

Alternate view: $\left(\frac{n^3}{m^2}\right)^2=\frac{n^6}{m^4}$

Try It

Simplify each expression and write your answer with positive exponents only.

Zero Exponents

Rule: Zero Exponents

Any nonzero base raised to the 0 power equals 1.

$a^0=1\quad (a\ne 0)$

We can also understand zero exponents using division.

Example 4.2.9 — Why $a^0=1$

Explore: $\frac{5^3}{5^3}$

Solution

Using the Quotient Property:

$\frac{5^3}{5^3}=5^{3-3}=5^0$

But any nonzero number divided by itself equals 1, so

$\frac{5^3}{5^3}=1$

Therefore $5^0=1$ .

This definition keeps the exponent rules consistent when dividing powers.

Example 4.2.10 — Zero exponent

Simplify: $x^0$

Solution

$x^0=1$ (for $x\ne 0$ )

Example 4.2.11 — Be careful with negatives

Simplify: $-x^0$

Solution

The exponent applies only to $x$ : $x^0=1$ .

So $-x^0=-(1)=-1$ .

Example 4.2.12 — Entire base vs. one factor

Simplify: $(3y)^0$

Solution

The entire base is $(3y)$ , so $(3y)^0=1$ (when $3y\ne 0$ ).

Example 4.2.13 — One factor has exponent 0

Simplify: $3y^0$

Solution

$3y^0=3\cdot 1=3$

Example 4.2.14 — A whole expression to the zero power

Simplify: $(4a^3b^2)^0$

Solution

$(4a^3b^2)^0=1$ (as long as $4a^3b^2\ne 0$ )

Try It

Applying Multiple Properties

When simplifying expressions, you may need to combine several exponent rules in one problem.

Simplify step by step, and write final answers with positive exponents.

Example 4.2.19 — Combine product and negative exponents

Simplify: $(p^{-2}q^3)(p^5q^{-1})$

Solution

$(p^{-2}q^3)(p^5q^{-1})=p^{-2+5}q^{3+(-1)}=p^3q^2$

Example 4.2.20 — Power of a product with negative exponent

Simplify: $(4k^{-3}m^2)^2$

Solution

$(4k^{-3}m^2)^2=4^2k^{-6}m^4=16k^{-6}m^4=\frac{16m^4}{k^6}$

Example 4.2.21 — Quotient with negative exponents

Simplify: $\frac{m^{-4}n^2}{m^{-2}n^5}$

Solution

$\frac{m^{-4}n^2}{m^{-2}n^5}=m^{-4-(-2)}n^{2-5}=m^{-2}n^{-3}=\frac{1}{m^2n^3}$

Example 4.2.22 — Power of a quotient with negative exponent in denominator

Simplify: $\left(\frac{2x^3y^2}{3z^{-2}}\right)^2$

Solution

First rewrite $z^{-2}$ in the denominator:

$\frac{2x^3y^2}{3z^{-2}}=\frac{2x^3y^2z^2}{3}$

Now square:

$\left(\frac{2x^3y^2z^2}{3}\right)^2=\frac{2^2x^6y^4z^4}{3^2}=\frac{4x^6y^4z^4}{9}$

Try It

Simplify each. Write your answer with positive exponents only.

Scientific Notation

Scientific notation provides a convenient way to write very large or very small numbers using powers of 10.

Definition

Scientific notation is written in the form

$a\times 10^n$

where $1\le a<10$ and $n$ is an integer.

Example 4.2.16 — Convert to scientific notation

Convert to scientific notation: 42,000 and 5,600,000

Solution

42,000 = $4.2\times 10^4$ (decimal moved 4 places left)

5,600,000 = $5.6\times 10^6$

Example 4.2.17 — Convert small numbers to scientific notation

Convert to scientific notation: 0.00056 and 0.0098

Solution

0.00056 = $5.6\times 10^{-4}$ (decimal moved 4 places right)

0.0098 = $9.8\times 10^{-3}$

Example 4.2.18 — Convert to standard form

Convert to standard form: $3.2\times 10^4$ and $7.5\times 10^{-3}$

Solution

$3.2\times 10^4$ = 32,000

$7.5\times 10^{-3}$ = 0.0075

Try It

Convert to scientific notation:

Try It

Convert to standard notation:

Concept Questions

Key Takeaways

Quotient Property: $\frac{a^m}{a^n}=a^{m-n}$
Negative exponents mean the base belongs in the denominator: $a^{-n}=\frac{1}{a^n}$ .
Zero exponents: $a^0=1$ (for $a\ne 0$ ).
Scientific notation expresses large or small numbers using powers of 10: $a\times 10^n$ .

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Essentials of Mathematics Copyright © 2026 by Scott McDaniel is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License, except where otherwise noted.

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