Lesson 2: Use Multiplication Properties of Exponents

New Concept

Ready to master exponents? This lesson introduces the multiplication properties of exponents. You'll learn how to simplify expressions by combining terms with the same base, raising powers to powers, and distributing exponents over products.

What’s next

Now that you have the big picture, get ready for interactive examples and practice cards where you'll apply these properties step-by-step.

Property

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

Examples

• The expression $4^3$ means you multiply the base $4$ by itself $3$ times: $4 \cdot 4 \cdot 4 = 64$.

• The expression $(-3)^4$ means you multiply the base $-3$ by itself $4$ times: $(-3)(-3)(-3)(-3) = 81$.

Property

If $a$ is a real number, and $m$ and $n$ are counting numbers, then

To multiply with like bases, add the exponents.

Examples

• To simplify $x^4 \cdot x^7$, since the bases are the same, we add the exponents: $x^{4+7} = x^{11}$.

• To simplify $5^2 \cdot 5^3$, we keep the base $5$ and add the exponents, giving $5^{2+3} = 5^5$, which equals $3125$.

Property

If $a$ is a real number and $m, n$ are whole numbers, then

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

Examples

• To simplify $(x^4)^3$, we keep the base $x$ and multiply the exponents: $x^{4 \cdot 3} = x^{12}$.

• To simplify $(2^3)^2$, we use the power property to get $2^{3 \cdot 2} = 2^6$, which equals $64$.

Property

If $a$ and $b$ are real numbers and $m$ is a whole number, then

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

Examples

• To simplify $(3y)^2$, the exponent $2$ applies to both the $3$ and the $y$, resulting in $3^2 y^2 = 9y^2$.

• To simplify $(-2ab)^3$, the exponent applies to $-2$, $a$, and $b$: $(-2)^3 a^3 b^3 = -8a^3b^3$.

Property

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

Examples

• To simplify $(x^2)^5 (x^4)^3$, first use the Power Property to get $x^{10} \cdot x^{12}$. Then use the Product Property to get $x^{10+12} = x^{22}$.

• To simplify $(-3a^2b^3)^2$, use the Product to a Power Property: $(-3)^2 (a^2)^2 (b^3)^2$. This simplifies to $9a^4b^6$.

Property

Since a monomial is an algebraic expression, we can use the properties of exponents to multiply monomials. To do this, first multiply the coefficients (the numbers in front), and then use the Product Property to add the exponents of like variables.

Examples

• To multiply $(5x^4)(3x^2)$, multiply the coefficients $5 \cdot 3 = 15$ and add the exponents for $x$: $x^{4+2} = x^6$. The result is $15x^6$.

• To multiply $(-2a^2b)(7ab^3)$, multiply coefficients $-2 \cdot 7 = -14$. For the variables, $a^{2+1} = a^3$ and $b^{1+3}=b^4$. The result is $-14a^3b^4$.