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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*}xxxxxxxxxyyyyyxxxx

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

Watch This

James Sousa: Exponential Notation

Guidance

In the expression \begin{align*}x^3\end{align*}x3, the \begin{align*}x\end{align*}x is called the base and the \begin{align*}3\end{align*}3 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*}x3=xxx
  • \begin{align*}2^4=2\cdot 2 \cdot 2 \cdot 2=16\end{align*}24=2222=16.

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.

RULE: 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*}
am×an=(a×a××a) (a×a××a)  m factorsn factorsam×an=(a×a×a×a) m+n factorsam×an=am+n

RULE: 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*}
(ab)n=(ab)×(ab)××(ab)n factors(ab)n=(a×a××a)×(b×b××b) n factors n factors(ab)n=anbn

Example A

Simplify \begin{align*}3^2 \times 3^3\end{align*}32×33.

Solution:

\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*}

32×3332+335The base is 3.Keep the base of 3 and add the exponents.This answer is in exponential form.

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*}

35=3×3×3×3×335=24332×33=35=243

Example B

Simplify \begin{align*}(x^3) (x^6)\end{align*}(x3)(x6).

Solution:

\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*}

(x3)(x6)x3+6x9(x3)(x6)=x9The base is x.Keep the base of x and add the exponents.The answer is in exponential form.

Example C

Simplify \begin{align*}y^5 \cdot y^2\end{align*}y5y2.

Solution:

\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*}

y5y2y5+2y7y5y2=y7The base is y.Keep the base of y and add the exponents.The answer is in exponential form.

Example D

Simplify \begin{align*}5x^2 y^3 \cdot 3xy^2\end{align*}5x2y33xy2.

Solution:

\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*}

5x2y33xy215(x2y3)(xy2)15x2+1y3+215x3y55x2y33xy2=15x3y5The bases are x and y.Multiply the coefficients - 5×3=15. Keep the base of x and y and addthe exponents of the same base. If a base does not have a writtenexponent, it is understood as 1.The answer is in exponential form.

Concept Problem Revisited

\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*}xxxxxxxxxyyyyyxxxx can be rewritten as \begin{align*} x^9 y^5 x^4\end{align*}x9y5x4. Then, you can use the rules of exponents to simplify the expression to \begin{align*}x^{13} y^5\end{align*}x13y5. This is certainly much quicker to write!

Vocabulary

Base
In an algebraic expression, the base is the variable, number, product or quotient, to which the exponent refers. Some examples are: In the expression \begin{align*}2^5\end{align*}25, ‘2’ is the base. In the expression \begin{align*}(-3y)^4\end{align*}(3y)4, ‘\begin{align*}-3y\end{align*}3y’ is the base.
Exponent
In an algebraic expression, the exponent is the number to the upper right of the base that tells how many times to multiply the base times itself. Some examples are:
In the expression \begin{align*}2^5\end{align*}25, ‘5’ is the exponent. It means to multiply 2 times itself 5 times as shown here: \begin{align*}2^5=2 \times 2 \times 2 \times 2 \times 2\end{align*}25=2×2×2×2×2.
In the expression \begin{align*}(-3y)^4\end{align*}(3y)4, ‘4’ is the exponent. It means to multiply \begin{align*}-3y\end{align*}3y times itself 4 times as shown here: \begin{align*}(-3y)^4=-3y \times -3y \times -3y \times -3y\end{align*}(3y)4=3y×3y×3y×3y.
Laws of Exponents
The laws of exponents are the algebra rules and formulas that tell us the operation to perform on the exponents when dealing with exponential expressions.

Guided Practice

Simplify each of the following expressions.

1. \begin{align*}(-3x)^2\end{align*}(3x)2

2. \begin{align*}(5xy)^3\end{align*}(5xy)3

3. \begin{align*}(2^3 \cdot 3^2)^2\end{align*}(2332)2

Answers:

1. \begin{align*}9x^2\end{align*}9x2. Here are the steps:

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

(3x)2=(3)2(x)2=9x2

2. \begin{align*}125x^3y^3\end{align*}125x3y3. Here are the steps:

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

(5x2y4)3=(5)3(x)3(y)3=125x3y3

3. \begin{align*}5184\end{align*}5184. Here are the steps:

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

(2332)2=(89)2=(72)2=5184

OR

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

(2332)2=(89)2=8292=6481=5184

Practice

Simplify each of the following expressions, if possible.

  1. \begin{align*}4^2\times 4^4\end{align*}42×44
  2. \begin{align*}x^4\cdot x^{12}\end{align*}x4x12
  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.

Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 6.1. 

Vocabulary

Base

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

Exponent

Exponents are used to describe the number of times that a term is multiplied by itself.
Power

Power

The "power" refers to the value of the exponent. For example, 3^4 is "three to the fourth power".

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