Use Multiplication Properties of Exponents

By the end of this section, you will be able to: simplify expressions with exponents; simplify expressions using the Product Property for Exponents; simplify expressions using the Power Property for Exponents; simplify expressions using the Product to a Power Property; simplify expressions by applying several properties; and multiply monomials.

Simplify Expressions with Exponents

Remember that an exponent indicates repeated multiplication of the same quantity. For example, $2^4$ means to multiply $2$ by itself $4$ times, so $2^4$ means $2 \cdot 2 \cdot 2 \cdot 2$ .

Let’s review the vocabulary for expressions with exponents.

Exponential notation. For any real number $a$ and counting number $m$ ,

a^m = \underbrace{a \cdot a \cdot a \cdots a}_{m \text{ factors}}

This is read $a$ to the $m^{\text{th}}$ power. In the expression $a^m$ , the exponent $m$ tells us how many times we use the base $a$ as a factor.

For example, $4^3$ means $4 \cdot 4 \cdot 4$ (three factors), and $(-9)^5$ means $(-9)(-9)(-9)(-9)(-9)$ (five factors).

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

Example. Simplify: (a) $4^3$ (b) $7^1$ (c) $\left(\tfrac{5}{6}\right)^2$ (d) $(0.63)^2$ .

(a) $4^3$ means multiply three factors of $4$ : $4 \cdot 4 \cdot 4 = 64$ .

(b) $7^1$ means multiply one factor of $7$ : $7^1 = 7$ .

(c) $\left(\tfrac{5}{6}\right)^2$ means multiply two factors: $\left(\tfrac{5}{6}\right)\left(\tfrac{5}{6}\right) = \tfrac{25}{36}$ .

(d) $(0.63)^2$ means multiply two factors: $(0.63)(0.63) = 0.3969$ .

Simplify: $6^3$ .

Simplify: $\tfrac{2}{5}^3$ .

Example. Simplify: (a) $(-5)^4$ (b) $-5^4$ .

(a) $(-5)^4$ means multiply four factors of $-5$ : $(-5)(-5)(-5)(-5) = 625$ .

(b) $-5^4$ means the opposite of $5^4$ , so we multiply four factors of $5$ and then take the opposite: $-(5 \cdot 5 \cdot 5 \cdot 5) = -625$ .

Notice the similarities and differences in these two parts. Why are the answers different? As we follow the order of operations, in part (a) the parentheses tell us to raise the $-5$ to the $4^{\text{th}}$ power. In part (b) we raise just the $5$ to the $4^{\text{th}}$ power and then take the opposite.

Simplify: $(-3)^4$ .

Simplify: $-3^4$ .

Simplify Expressions Using the Product Property for 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 exponents 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. First, we will look at an example that leads to the Product Property. Consider $x^2 \cdot x^3$ . What does this mean?

x^2 \cdot x^3 = \underbrace{(x \cdot x)}_{2 \text{ factors}} \cdot \underbrace{(x \cdot x \cdot x)}_{3 \text{ factors}} = \underbrace{x \cdot x \cdot x \cdot x \cdot x}_{5 \text{ factors}} = 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.

Product Property for Exponents. If $a$ is a real number and $m$ and $n$ are counting numbers, then

a^m \cdot a^n = a^{m+n}

To multiply with like bases, add the exponents.

An example with numbers helps to verify this property: $2^2 \cdot 2^3 \overset{?}{=} 2^{2+3}$ , so $4 \cdot 8 \overset{?}{=} 2^5$ , giving $32 = 32$ ✓.

Example. Simplify: $y^5 \cdot y^6$ .

Use the product property, $a^m \cdot a^n = a^{m+n}$ , to add the exponents: $y^5 \cdot y^6 = y^{5+6} = y^{11}$ .

Simplify: $b^9 \cdot b^8$ .

Example. Simplify: (a) $2^5 \cdot 2^9$ (b) $3 \cdot 3^4$ .

(a) The bases are the same, so add the exponents: $2^5 \cdot 2^9 = 2^{5+9} = 2^{14}$ .

(b) Write $3$ as $3^1$ , then add the exponents: $3 \cdot 3^4 = 3^1 \cdot 3^4 = 3^{1+4} = 3^5$ .

Simplify: $5 \cdot 5^5$ . Write the answer as a power of 5.

Simplify: $7^6 \cdot 7^8$ . Write the answer as a power of 7.

Example. Simplify: (a) $a^7 \cdot a$ (b) $x^{27} \cdot x^{13}$ .

(a) Rewrite $a$ as $a^1$ , then use the product property: $a^7 \cdot a = a^7 \cdot a^1 = a^{7+1} = a^8$ .

(b) The bases are the same, so add the exponents: $x^{27} \cdot x^{13} = x^{27+13} = x^{40}$ .

Simplify: $p^5 \cdot p$ . Write the answer as a power of p.

Simplify: $y^{14} \cdot y^{29}$ . Write the answer as a power of y.

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

Example. Simplify: $d^4 \cdot d^5 \cdot d^2$ .

Add the exponents, since the bases are the same: $d^4 \cdot d^5 \cdot d^2 = d^{4+5+2} = d^{11}$ .

Simplify: $x^6 \cdot x^4 \cdot x^8$ . Write the answer as a power of x.

Simplify Expressions Using the Power Property for Exponents

Now let’s look at an exponential expression that contains a power raised to a power. Consider $\left(x^2\right)^3$ . What does this mean?

\left(x^2\right)^3 = \underbrace{x^2 \cdot x^2 \cdot x^2}_{3 \text{ factors}} = x^{2+2+2} = x^6

Notice that $6$ is the product of the exponents, $2$ and $3$ . We multiplied the exponents. This leads to the Power Property for Exponents.

Power Property for Exponents. If $a$ is a real number and $m$ and $n$ are whole numbers, then

\left(a^m\right)^n = a^{m \cdot n}

To raise a power to a power, multiply the exponents.

An example with numbers helps to verify this property: $\left(3^2\right)^3 \overset{?}{=} 3^{2 \cdot 3}$ , so $(9)^3 \overset{?}{=} 3^6$ , giving $729 = 729$ ✓.

Example. Simplify: (a) $\left(y^5\right)^9$ (b) $\left(4^4\right)^7$ .

(a) Use the power property, $\left(a^m\right)^n = a^{m \cdot n}$ , to multiply the exponents: $\left(y^5\right)^9 = y^{5 \cdot 9} = y^{45}$ .

(b) Multiply the exponents: $\left(4^4\right)^7 = 4^{4 \cdot 7} = 4^{28}$ .

Simplify: $(b^7)^5$ . Write the answer as a power of b.

Simplify: $(z^6)^9$ . Write the answer as a power of z.

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 $(2x)^3$ . What does this mean?

(2x)^3 = (2x)(2x)(2x) = \underbrace{(2 \cdot 2 \cdot 2)}_{3 \text{ factors of } 2} \cdot \underbrace{(x \cdot x \cdot x)}_{3 \text{ factors of } x} = 2^3 \cdot x^3

Notice that each factor was raised to the power, so $(2x)^3$ is $2^3 \cdot x^3$ . The exponent applies to each of the factors! This leads to the Product to a Power Property for Exponents.

Product to a Power Property for Exponents. If $a$ and $b$ are real numbers and $m$ is a whole number, then

(ab)^m = a^m b^m

To raise a product to a power, raise each factor to that power.

An example with numbers helps to verify this property: $(2 \cdot 3)^2 \overset{?}{=} 2^2 \cdot 3^2$ , so $6^2 \overset{?}{=} 4 \cdot 9$ , giving $36 = 36$ ✓.

Example. Simplify: (a) $(-9d)^2$ (b) $(3mn)^3$ .

(a) Use the Product to a Power Property, $(ab)^m = a^m b^m$ , then simplify: $(-9d)^2 = (-9)^2 d^2 = 81d^2$ .

(b) Raise each factor to the third power: $(3mn)^3 = (3)^3 m^3 n^3 = 27m^3 n^3$ .

Simplify: $(-12y)^2$ .

Simplify: (2wx)^5.

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.

Properties of Exponents. If $a$ and $b$ are real numbers, and $m$ and $n$ are whole numbers, then

Product Property: $a^m \cdot a^n = a^{m+n}$
Power Property: $\left(a^m\right)^n = a^{m \cdot n}$
Product to a Power: $(ab)^m = a^m b^m$

All exponent properties hold true for any real numbers $m$ and $n$ . Right now, we only use whole number exponents.

Example. Simplify: (a) $\left(y^3\right)^6 \left(y^5\right)^4$ (b) $\left(-6x^4 y^5\right)^2$ .

(a) Use the Power Property on each factor, then add the exponents: $\left(y^3\right)^6 \left(y^5\right)^4 = y^{18} \cdot y^{20} = y^{38}$ .

(b) Use the Product to a Power Property, then the Power Property, then simplify:

\left(-6x^4 y^5\right)^2 = (-6)^2 \left(x^4\right)^2 \left(y^5\right)^2 = 36 x^8 y^{10}

Simplify: $(a^4)^5 (a^7)^4$ . Write the answer as a power of a.

Simplify: $(-2c^4 d^2)^3$ .

Example. Simplify: (a) $(5m)^2 \left(3m^3\right)$ (b) $\left(3x^2 y\right)^4 \left(2xy^2\right)^3$ .

(a) Raise $5m$ to the second power, then rearrange and multiply the constants while adding the exponents:

(5m)^2 \left(3m^3\right) = 25m^2 \cdot 3m^3 = 25 \cdot 3 \cdot m^2 \cdot m^3 = 75m^5

(b) Use the Product to a Power Property on each factor, then rearrange and combine:

\left(3x^2 y\right)^4 \left(2xy^2\right)^3 = \left(81x^8 y^4\right)\left(8x^3 y^6\right) = 81 \cdot 8 \cdot x^8 \cdot x^3 \cdot y^4 \cdot y^6 = 648 x^{11} y^{10}

Simplify: $(5n)^2 (3n^{10})$ .

Simplify: (c^4 d^2)^5 (3cd^5)^4.

Multiply Monomials

Since a monomial is an algebraic expression, we can use the properties of exponents to multiply monomials.

Example. Multiply: $\left(3x^2\right)\left(-4x^3\right)$ .

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

\left(3x^2\right)\left(-4x^3\right) = 3 \cdot (-4) \cdot x^2 \cdot x^3 = -12x^5

Multiply: $(5y^7)(-7y^4)$ .

Multiply: $(-6b^4)(-9b^5)$ .

Example. Multiply: $\left(\tfrac{5}{6}x^3 y\right)\left(12xy^2\right)$ .

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

\left(\tfrac{5}{6}x^3 y\right)\left(12xy^2\right) = \tfrac{5}{6} \cdot 12 \cdot x^3 \cdot x \cdot y \cdot y^2 = 10 x^4 y^3

Multiply: $(\tfrac{2}{5} a^4 b^3)(15ab^3)$ .

Multiply: $(\tfrac{2}{3} r^5 s)(12r^6 s^7)$ .

Key terms

exponential notation — $a^m$ means multiply $m$ factors of $a$ ; the base $a$ is used as a factor $m$ times. Product Property for Exponents — to multiply powers with like bases, add the exponents: $a^m \cdot a^n = a^{m+n}$ . Power Property for Exponents — to raise a power to a power, multiply the exponents: $\left(a^m\right)^n = a^{m \cdot n}$ . Product to a Power Property for Exponents — to raise a product to a power, raise each factor to that power: $(ab)^m = a^m b^m$ . monomial — a one-term algebraic expression, such as $-12x^5$ , that we can multiply by applying these properties of exponents.

This section is adapted from Elementary Algebra 2e, Section 6.2: 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: recast the worked-example step tables as prose and typeset equations; omitted the Be Prepared quiz, Self Check checklist, media links, and end-of-section exercises; and converted the practice problems (“Try Its”) into interactive exercises with instant feedback.

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