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Exponential Properties Involving Products

Add exponents to multiply exponents by other exponents

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Product Rules for Exponents

Suppose you have the expression:

\begin{align*}x\cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot y \cdot y \cdot y \cdot y \cdot y \cdot x \cdot x \cdot x \cdot x\end{align*}

How could you write this expression in a more concise way? 

Product Rules for Exponents

In the expression \begin{align*}x^3\end{align*}, the \begin{align*}x\end{align*} is called the base and the \begin{align*}3\end{align*} is called the exponent. Exponents are often referred to as powers. When an exponent is a positive whole number, it tells you how many times to multiply the base by itself. For example:

  • \begin{align*}x^3=x\cdot x \cdot x\end{align*}
  • \begin{align*}2^4=2\cdot 2 \cdot 2 \cdot 2=16\end{align*}.

There are many rules that have to do with exponents (often called the Laws of Exponents) that are helpful to know so that you can work with expressions and equations that involve exponents more easily. Here you will learn two rules that have to do with exponents and products.

  1. To multiply two terms with the same base, add the exponents.
\begin{align*}& a^m \times a^n = \underleftrightarrow{(a \times a \times \ldots \times a)} \ \underleftrightarrow{(a \times a \times \ldots \times a)}\\ & \qquad \qquad \qquad \qquad \ {\color{red}\downarrow} \qquad \qquad \qquad \quad \ {\color{red}\downarrow}\\ & \qquad \qquad \qquad {\color{red} m \ \text{factors}} \qquad \qquad {\color{red} n \ \text{factors}}\\ & a^m \times a^n = \underleftrightarrow{(a \times a \times a \ldots \times a)}\\ & \qquad \qquad \qquad \qquad \ {\color{red}\downarrow}\\ & \qquad \qquad \qquad {\color{red} m+n \ \text{factors}}\\ & a^m \times a^n=a^{{\color{red}m+n}}\end{align*}
  1. To raise a product to a power, raise each of the factors to the power.
\begin{align*}&(ab)^n=\underleftrightarrow{(ab) \times (ab) \times \ldots \times (ab)}\\ & \qquad \qquad \qquad \qquad {\color{red}\downarrow}\\ & \qquad \qquad \qquad {\color{red}n} \ {\color{red}\text{factors}}\\ & (ab)^n=\underleftrightarrow{(a \times a \times \ldots \times a)} \times \underleftrightarrow{(b \times b \times \ldots \times b)}\\ & \qquad \qquad \qquad \quad {\color{red}\downarrow} \qquad \qquad \qquad \qquad {\color{red}\downarrow}\\ & \qquad \qquad \quad \ {\color{red}n} \ {\color{red}\text{factors}} \qquad \qquad \ {\color{red}n} \ {\color{red}\text{factors}}\\ & (ab)^n=a^{{\color{red}n}} b^{{\color{red}n}}\end{align*}

Let's simplify the following expressions:

  1. \begin{align*}3^2 \times 3^3\end{align*}

\begin{align*}& 3^2 \times 3^3 && \text{The base is} \ 3.\\ & 3^{2+3} && \text{Keep the base of} \ 3 \ \text{and add the exponents.}\\ & 3^{\color{red}5} && \text{This answer is in exponential form.}\end{align*}

The answer can be taken one step further. The base is numerical so the term can be evaluated.

\begin{align*}& 3^5=3 \times 3 \times 3 \times 3 \times 3\\ & {\color{red}3^5}={\color{red}243}\\ & \boxed{3^2 \times 3^3 = 3^5=243}\end{align*}

  1. \begin{align*}(x^3) (x^6)\end{align*}.

\begin{align*}& (x^3)(x^6) && \text{The base is} \ x.\\ & x^{3+6} && \text{Keep the base of} \ x \ \text{and add the exponents.}\\ & x^{\color{red}9} && \text{The answer is in exponential form.}\\ & \boxed{(x^3)(x^6)=x^9}\end{align*}

  1. \begin{align*}y^5 \cdot y^2\end{align*}.

\begin{align*}& y^5 \cdot y^2 && \text{The base is} \ y.\\ & y^{5+2} && \text{Keep the base of} \ y \ \text{and add the exponents.}\\ & y^{\color{red}7} && \text{The answer is in exponential form.}\\ & \boxed{y^5 \cdot y^2=y^7}\end{align*}

  1. \begin{align*}5x^2 y^3 \cdot 3xy^2\end{align*}.

\begin{align*}& 5x^2 y^3 \cdot 3xy^2 && \text{The bases are} \ x \ \text{and} \ y.\\ & 15(x^2 y^3)(xy^2) && \text{Multiply the coefficients -} \ 5 \times 3=15. \ \text{Keep the base of} \ x \ \text{and} \ y \ \text{and add}\\ & && \text{the exponents of the same base. If a base does not have a written}\\ & && \text{exponent, it is understood as} \ 1.\\ & 15x^{2+1} y^{3+2}\\ & 15x^{\color{red}3} y^{\color{red}5} && \text{The answer is in exponential form.}\\ & \boxed{5x^2 y^3 \cdot 3xy^2=15x^3y^5}\end{align*}


Example 1

Earlier, you were asked to write the following expression in a more concise way;

\begin{align*}x\cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot y \cdot y \cdot y \cdot y \cdot y \cdot x \cdot x \cdot x \cdot x\end{align*} 

The expression can be rewritten as \begin{align*} x^9 y^5 x^4\end{align*}. Then, you can use the rules of exponents to simplify the expression to \begin{align*}x^{13} y^5\end{align*}. This is certainly much quicker to write!

Example 2

Simplify the following expression:


 \begin{align*}9x^2\end{align*}. Here are the steps:

\begin{align*}(-3x)^2&=(-3)^2\cdot(x)^2\\ &=9x^2\end{align*}

Example 3

Simplify the following expression: 


\begin{align*}125x^3y^3\end{align*}. Here are the steps:

\begin{align*}(5x^2 y^4)^3&=(5)^3\cdot (x)^3\cdot (y)^3\\ &=125x^3y^3\end{align*}

Example 4

Simplify the following expression:

\begin{align*}(2^3 \cdot 3^2)^2\end{align*}

Here are the steps:

\begin{align*}(2^3 \cdot 3^2)^2&=(8\cdot 9)^2\\ &=(72)^2\\ &=5184\end{align*}


\begin{align*}(2^3 \cdot 3^2)^2&=(8\cdot 9)^2\\ &=8^2\cdot 9^2 \\ &=64\cdot 81\\ &=5184\end{align*}


Simplify each of the following expressions, if possible.

  1. \begin{align*}4^2\times 4^4\end{align*}
  2. \begin{align*}x^4\cdot x^{12}\end{align*}
  3. \begin{align*}(3x^2y^4)(9xy^5z)\end{align*}
  4. \begin{align*}(2xy)^2(4x^2y^3)\end{align*}
  5. \begin{align*}(3x)^5(2x)^2(3x^4)\end{align*}
  6. \begin{align*}x^3y^2z\cdot 4xy^2z^7\end{align*}
  7. \begin{align*}x^2y^3+xy^2\end{align*}
  8. \begin{align*}(0.1xy)^{4}\end{align*}
  9. \begin{align*}(xyz)^6\end{align*}
  10. \begin{align*}2x^4(x^2-y^2)\end{align*}
  11. \begin{align*}3x^5-x^2\end{align*}
  12. \begin{align*}3x^8(x^2-y^4)\end{align*}

Expand and then simplify each of the following expressions.

  1. \begin{align*}(x^5)^3\end{align*}
  2. \begin{align*}(x^6)^8\end{align*}
  3. \begin{align*}(x^a)^b\end{align*} Hint: Look for a pattern in the previous two problems.

Review (Answers)

To see the Review answers, open this PDF file and look for section 6.1. 

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Base When a value is raised to a power, the value is referred to as the base, and the power is called the exponent. In the expression 32^4, 32 is the base, and 4 is the exponent.
Exponent Exponents are used to describe the number of times that a term is multiplied by itself.
Power The "power" refers to the value of the exponent. For example, 3^4 is "three to the fourth power".

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