Use Multiplication Properties of Exponents

Simplify expressions with exponents

Remember that an exponent indicates repeated multiplication of the same quantity. For example, $2^4$ means to multiply four factors of $2$, so $2^4$ means $2 \cdot 2 \cdot 2 \cdot 2$. This format is known as exponential notation.

In the expression $a^m$, the exponent tells us how many times we use the base $a$ as a factor.

Before we begin working with variable expressions containing exponents, let’s simplify a few expressions involving only numbers.

Example. Simplify: (a) $5^3$ (b) $9^1$.

(a) Multiply $3$ factors of $5$: $5^3 = 5 \cdot 5 \cdot 5 = 125$.

(b) Multiply $1$ factor of $9$: $9^1 = 9$.

Example. Simplify: (a) $\left(\tfrac{7}{8}\right)^2$ (b) $(0.74)^2$.

(a) Multiply two factors: $\left(\tfrac{7}{8}\right)^2 = \left(\tfrac{7}{8}\right)\left(\tfrac{7}{8}\right) = \tfrac{49}{64}$.

(b) Multiply two factors: $(0.74)^2 = (0.74)(0.74) = 0.5476$.

Example. Simplify: (a) $(-3)^4$ (b) $-3^4$.

(a) Multiply four factors of $-3$: $(-3)^4 = (-3)(-3)(-3)(-3) = 81$.

(b) Here the exponent applies only to $3$, and the negative sign out front is separate: $-3^4 = -(3 \cdot 3 \cdot 3 \cdot 3) = -81$.

Notice the similarities and differences in parts (a) and (b). Why are the answers different? In part (a) the parentheses tell us to raise the $-3$ to the $4$th power. In part (b) we raise only the $3$ to the $4$th power and then find the opposite.

Simplify expressions using the Product Property of Exponents

You have seen that when you combine like terms by adding and subtracting, you need to have the same base with the same exponent. But when you multiply and divide, the bases may be different, and sometimes the bases may be different, too. We’ll derive the properties of exponents by looking for patterns in several examples. All the exponent properties hold true for any real numbers, but right now we will only use whole number exponents.

First, we will look at an example that leads to the Product Property. Consider $x^2 \cdot x^3$. This means $x \cdot x$ (2 factors) times $x \cdot x \cdot x$ (3 factors), or $5$ factors of $x$ altogether, so $x^2 \cdot x^3 = x^5$. Notice that $5$ is the sum of the exponents, $2$ and $3$:

The base stayed the same and we added the exponents. This leads to the Product Property for Exponents.

An example with numbers helps to verify this property:

Example. Simplify: $x^5 \cdot x^7$.

Use the Product Property, $a^m \cdot a^n = a^{m+n}$:

Example. Simplify: $b^4 \cdot b$.

Rewrite $b$ as $b^1$, then use the Product Property:

Example. Simplify: $2^7 \cdot 2^9$.

Use the Product Property, $a^m \cdot a^n = a^{m+n}$:

Example. Simplify: $y^{17} \cdot y^{23}$.

The bases are the same, so add the exponents:

We can extend the Product Property of Exponents to more than two factors.

Example. Simplify: $x^3 \cdot x^4 \cdot x^2$.

Add the exponents, since the bases are the same:

Simplify expressions using the Power Property of Exponents

Now let’s look at an exponential expression that contains a power raised to a power. Consider $(x^2)^3$. This means $x^2 \cdot x^2 \cdot x^2$, and each factor of $x^2$ is itself $x \cdot x$, so there are $6$ factors of $x$ altogether:

Notice that $6$ is the product of the exponents, $2$ and $3$: $(x^2)^3$ is $x^{2 \cdot 3}$, or $x^6$. We multiplied the exponents. This leads to the Power Property for Exponents.

An example with numbers helps to verify this property:

Example. Simplify: (a) $(x^5)^7$ (b) $(3^6)^8$.

(a) Use the Power Property, $(a^m)^n = a^{m \cdot n}$: $(x^5)^7 = x^{5 \cdot 7} = x^{35}$.

(b) Use the Power Property: $(3^6)^8 = 3^{6 \cdot 8} = 3^{48}$.

Simplify expressions using the Product to a Power Property

We will now look at an expression containing a product that is raised to a power. Consider $(2x)^3$. This means $2x \cdot 2x \cdot 2x$. If we group the like factors together, we get $2 \cdot 2 \cdot 2 \cdot x \cdot x \cdot x$, which is $2^3 \cdot x^3$. Notice that each factor was raised to the power:

The exponent applies to each of the factors. This leads to the Product to a Power Property for Exponents.

An example with numbers helps to verify this property:

Example. Simplify: $(-11x)^2$.

Use the Power of a Product Property, $(ab)^m = a^m b^m$:

Example. Simplify: $(3xy)^3$.

Raise each factor to the third power, then simplify:

Simplify expressions by applying several properties

We now have three properties for multiplying expressions with exponents. Let’s summarize them and then do some examples that use more than one of the properties.

Example. Simplify: $(x^2)^6 (x^5)^4$.

Use the Power Property, then add the exponents since the bases match:

Example. Simplify: $(-7x^3y^4)^2$.

Take each factor to the second power, then use the Power Property:

Example. Simplify: $(6n)^2(4n^3)$.

Raise $6n$ to the second power, simplify, then use the Commutative Property to group the constants and like bases together, and finally multiply the constants and add the exponents:

Notice that in the first monomial, the exponent was outside the parentheses and it applied to both factors inside. In the second monomial, the exponent was inside the parentheses and so it only applied to the $n$.

Example. Simplify: $(3p^2q)^4(2pq^2)^3$.

Use the Power of a Product Property on each factor, then the Commutative Property to group like bases, and finally multiply the constants and add the exponents for each variable:

Multiply monomials

Since a monomial is an algebraic expression, we can use the properties for simplifying expressions with exponents to multiply monomials.

Example. Multiply: $(4x^2)(-5x^3)$.

Use the Commutative Property to rearrange the factors, then multiply:

Example. Multiply: $\left(\tfrac{3}{4}c^3d\right)(12cd^2)$.

Use the Commutative Property to rearrange the factors, then multiply:

Key terms

exponential notation — writing $a^m$ to mean $m$ factors of $a$ multiplied together. Product Property of Exponents — to multiply expressions with the same base, add the exponents: $a^m \cdot a^n = a^{m+n}$. Power Property of Exponents — to raise a power to a power, multiply the exponents: $(a^m)^n = a^{m \cdot n}$. Product to a Power Property of Exponents — to raise a product to a power, raise each factor to that power: $(ab)^m = a^m b^m$. monomial — an algebraic expression with one term, such as a single number, variable, or product of numbers and variables with whole-number exponents.

This section is adapted from Prealgebra 2e, Section 10.2: Use Multiplication Properties of Exponents by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: omitted the Be Prepared quiz, Self Check checklist, Media links, and end-of-section exercises; condensed the pattern-building tables that derive each property into short prose descriptions; and converted the practice problems (“Try Its”) into interactive exercises with instant feedback.