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Exponential Properties Involving Products

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Practice Exponential Properties Involving Products
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Product Rules for Exponents

Suppose you have the expression:

$x\cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot y \cdot y \cdot y \cdot y \cdot y \cdot x \cdot x \cdot x \cdot x$

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

Guidance

In the expression $x^3$ , the $x$ is called the base and the $3$ 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:

• $x^3=x\cdot x \cdot x$
• $2^4=2\cdot 2 \cdot 2 \cdot 2=16$ .

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.

$& a^m \times a^n = \underleftrightarrow{(a \times a \times \ldots \times a)} \ \underleftrightarrow{(a \times a \times \ldots \times a)}\\& \qquad \qquad \qquad \qquad \ {\color{red}\downarrow} \qquad \qquad \qquad \quad \ {\color{red}\downarrow}\\& \qquad \qquad \qquad {\color{red} m \ \text{factors}} \qquad \qquad {\color{red} n \ \text{factors}}\\& a^m \times a^n = \underleftrightarrow{(a \times a \times a \ldots \times a)}\\& \qquad \qquad \qquad \qquad \ {\color{red}\downarrow}\\& \qquad \qquad \qquad {\color{red} m+n \ \text{factors}}\\& a^m \times a^n=a^{{\color{red}m+n}}$

RULE: To raise a product to a power, raise each of the factors to the power.

$&(ab)^n=\underleftrightarrow{(ab) \times (ab) \times \ldots \times (ab)}\\& \qquad \qquad \qquad \qquad {\color{red}\downarrow}\\& \qquad \qquad \qquad {\color{red}n} \ {\color{red}\text{factors}}\\& (ab)^n=\underleftrightarrow{(a \times a \times \ldots \times a)} \times \underleftrightarrow{(b \times b \times \ldots \times b)}\\& \qquad \qquad \qquad \quad {\color{red}\downarrow} \qquad \qquad \qquad \qquad {\color{red}\downarrow}\\& \qquad \qquad \quad \ {\color{red}n} \ {\color{red}\text{factors}} \qquad \qquad \ {\color{red}n} \ {\color{red}\text{factors}}\\& (ab)^n=a^{{\color{red}n}} b^{{\color{red}n}}$

Example A

Simplify $3^2 \times 3^3$ .

Solution:

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

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

$& 3^5=3 \times 3 \times 3 \times 3 \times 3\\& {\color{red}3^5}={\color{red}243}\\& \boxed{3^2 \times 3^3 = 3^5=243}$

Example B

Simplify $(x^3) (x^6)$ .

Solution:

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

Example C

Simplify $y^5 \cdot y^2$ .

Solution:

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

Example D

Simplify $5x^2 y^3 \cdot 3xy^2$ .

Solution:

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

Concept Problem Revisited

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

Guided Practice

Simplify each of the following expressions.

1. $(-3x)^2$

2. $(5xy)^3$

3. $(2^3 \cdot 3^2)^2$

1. $9x^2$ . Here are the steps:

$(-3x)^2&=(-3)^2\cdot(x)^2\\&=9x^2$

2. $125x^3y^3$ . Here are the steps:

$(5x^2 y^4)^3&=(5)^3\cdot (x)^3\cdot (y)^3\\&=125x^3y^3$

3. $5184$ . Here are the steps:

$(2^3 \cdot 3^2)^2&=(8\cdot 9)^2\\&=(72)^2\\&=5184$

OR

$(2^3 \cdot 3^2)^2&=(8\cdot 9)^2\\&=8^2\cdot 9^2 \\&=64\cdot 81\\&=5184$

Explore More

Simplify each of the following expressions, if possible.

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

Expand and then simplify each of the following expressions.

1. $(x^5)^3$
2. $(x^6)^8$
3. $(x^a)^b$ Hint: Look for a pattern in the previous two problems.

Vocabulary Language: English

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".