4.1 Properties of Exponents

Learning Objectives

After completing this section, you should be able to:
  1. Simplify expressions with exponents.
  2. Simplify expressions using the Product Property for Exponents.
  3. Simplify expressions using the Power Property for Exponents.
  4. Simplify expressions using the Power of a Product.
  5. Simplify expressions using the Power of a Quotient.
  6. Apply several properties together in one problem.

Simplifying Expressions with Exponents

Suppose a group of 10,000 people each earn an average of $100,000. What is the total amount earned?
We multiply 10{,}000 \cdot 100{,}000. Multiplying by 10’s is easy—we just “add the zeros.”
(4 zeros + 5 zeros = 9 zeros)
10{,}000 \cdot 100{,}000 = 1{,}000{,}000{,}000
Why does this work?
We are multiplying powers of 10. The exponent tells us how many zeros:
Table 4.1
10^1 = 10
10^2 = 100
10^3 = 1,000
10^4 = 10,000
10^5 = 100,000
So in our problem:
10^4 \cdot 10^5 = 10^9
When we multiply powers of 10, we simply add their exponents.
Recall: In a number like 10^2, the 10 is the base and the 2 is the exponent (or power).

Does this work with other bases?

Try it: 2^3 \cdot 2^4
2^3 = 2 \cdot 2 \cdot 2
2^4 = 2 \cdot 2 \cdot 2 \cdot 2
Together:
2^3 \cdot 2^4 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 2^7
Yes—it works for any base!

Product of Powers

Rule: Product of Powers

When multiplying powers with the same base, add the exponents.
a^m \cdot a^n = a^{m+n}

Example 4.1.1 — Product of powers

Simplify: x^3 \cdot x^5
Solution
x^3 \cdot x^5 = x^{3+5} = x^8

Example 4.1.2 — Product of powers with a coefficient

Simplify: 2y^2 \cdot y^6
Solution
2y^2 \cdot y^6 = 2y^{2+6} = 2y^8

Example 4.1.3 — Product of powers with grouping

Simplify: (4a^4)(a^2)
Solution
(4a^4)(a^2) = 4a^{4+2} = 4a^6

Note

The bases must be the same to apply this rule. For example, 4^3 \cdot 6^{20} cannot be combined because the bases differ.

Try It

Power of a Power

Rule: Power of a Power

When raising a power to another power, multiply the exponents.
(a^m)^n = a^{mn}

Example 4.1.4 — Power of a power

Simplify: (x^2)^3
Solution
(x^2)^3 = x^{2\cdot 3} = x^6

Example 4.1.5 — Power of a power (numbers)

Simplify: (7^3)^2
Solution
(7^3)^2 = 7^{3\cdot 2} = 7^6

Example 4.1.6 — Power of a power (variables)

Simplify: (a^4)^2
Solution
(a^4)^2 = a^{4\cdot 2} = a^8

Why does this work?

Raising a power to another power means repeating the same factor:
(7^3)^2 = (7^3)(7^3) = 7^{3+3} = 7^6

Try It

Power of a Product

Rule: Power of a Product

When raising a product to a power, apply the exponent to each factor inside the parentheses.
(ab)^m = a^m b^m

Example 4.1.7 — Power of a product

Simplify: (xy)^3
Solution
(xy)^3 = x^3y^3

Example 4.1.8 — Power of a product with a coefficient

Simplify: (2ab)^2
Solution
(2ab)^2 = 2^2a^2b^2 = 4a^2b^2

Example 4.1.9 — Power of a product with an exponent inside

Simplify: (3y^3)^2
Solution
(3y^3)^2 = 3^2(y^3)^2 = 9y^{3\cdot 2} = 9y^6

Try It

Power of a Quotient

Rule: Power of a Quotient

When raising a fraction to a power, apply the exponent to both numerator and denominator.
\left(\frac{a}{b}\right)^m=\frac{a^m}{b^m}

Example 4.1.10 — Power of a quotient

Simplify: \left(\frac{x}{y}\right)^2
Solution
\left(\frac{x}{y}\right)^2=\frac{x^2}{y^2}

Example 4.1.11 — Power of a quotient with a coefficient

Simplify: \left(\frac{3a}{b}\right)^3
Solution
\left(\frac{3a}{b}\right)^3=\frac{3^3a^3}{b^3}=\frac{27a^3}{b^3}

Example 4.1.12 — Power of a quotient (multiple factors)

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

Try It

Applying Multiple Properties

Note: The example numbering in this section matches the accompanying videos. Since the chapter was reorganized after the videos were recorded, some example numbers are not sequential.

 

 

Sometimes you’ll need to use several exponent rules in the same problem. Apply the power and product rules carefully, one step at a time.

Example 4.1.18 — Multiple properties

Simplify: (2x^2y^3)^2 \cdot x^3y
Solution
Step 1: Apply the Power of a Product Rule.
(2x^2y^3)^2 = 2^2x^{4}y^{6} = 4x^4y^6
Step 2: Multiply using the Product of Powers Rule.
4x^4y^6 \cdot x^3y = 4x^{4+3}y^{6+1} = 4x^7y^7
Final Answer: 4x^7y^7

Example 4.1.19 — Multiple properties

Simplify: (3a^2b^3)^2 \cdot (2a^3b)
Solution
Step 1: Apply the Power of a Product Rule.
(3a^2b^3)^2 = 3^2a^{4}b^{6} = 9a^4b^6
Step 2: Multiply using the Product of Powers Rule.
9a^4b^6 \cdot 2a^3b = 18a^{4+3}b^{6+1} = 18a^7b^7
Final Answer: 18a^7b^7

Example 4.1.20 — Power of a power + product

Simplify: \left((2m^2n^3)^2\right)^3
Solution
Step 1: Apply the Power of a Product Rule (inside).
(2m^2n^3)^2 = 2^2m^4n^6 = 4m^4n^6
Step 2: Apply the Power of a Power Rule.
(4m^4n^6)^3 = 4^3m^{12}n^{18} = 64m^{12}n^{18}
Final Answer: 64m^{12}n^{18}

Try It

Looking Ahead

In this section, we focused on exponent rules involving multiplication. In the next section, you’ll learn how to handle exponents when dividing powers (the Quotient Property), which introduces the idea of subtracting exponents.

Exercises

Concept Check

Key Takeaways

  • Add exponents when multiplying powers with the same base.
  • Multiply exponents when raising a power to another power.
  • Apply exponents to all factors in a product or quotient.
  • Simplify step by step — one property at a time.

 

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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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