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# Exponential Properties Involving Products

## Add exponents to multiply exponents by other exponents

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Practice Exponential Properties Involving Products
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Recognize and Apply the Power of a Product Property

Have you ever tried to square a monomial? Do you know how to do it? Take a look at this dilemma.

A square platform has a side length of $6a^2$ .

How can we find the area of the platform?

This Concept will show you how to use the Power of a Product with monomials. Then you will be able to find the area of the square platform.

### Guidance

When multiplying monomials, an exponent is applied to the constant, variable, or quantity that is directly to its left . However, we only applied exponents to single variables.

Exponents can also be applied to products using parentheses.

Look at this one.

$(5x)^4$

If we apply the exponent 4 to whatever is directly to its left, we would apply it to the parentheses, not just the $x$ . The parentheses are directly to the left of the 4. This indicates that the entire product in the parentheses is taken to the $4^{th}$ power. We can also write this in expanded form.

$& (5x)^4 \\&=(5x)(5x)(5x)(5x)$

Now we multiply the monomials as we have already learned—by placing like factors next to each other, multiplying the coefficients, and simplifying using exponents.

$&=5 \cdot 5 \cdot 5 \cdot 5 \cdot x \cdot x \cdot x \cdot x \\&=625x^4$

This is the Power of a Product Property which says, for any nonzero numbers $a$ and $b$ and any integer $n$

$(ab)^n=a^n b^n$

Here is another one.

$& (7h)^3 \\&=(7h)(7h)(7h) \\&=7 \cdot 7 \cdot 7 \cdot h \cdot h \cdot h \\&=343 h^3$

You can see that whether we have positive or negative integers or both, we can still use the Power of a Product Property. You may have already noticed a pattern with the exponents and the final product. When you multiply like bases, there is another shortcut—you can add the exponents of like bases. Another way of saying it is:

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

Take a look at this one.

$& (-2x^4)^5 \\&=(-2x^4)(-2x^4)(-2x^4)(-2x^4)(-2x^4) \\&=-2 \cdot -2 \cdot -2 \cdot -2 \cdot -2 \cdot x^4 \cdot x^4 \cdot x^4 \cdot x^4 \cdot x^4 \\&=-2 \cdot -2 \cdot -2 \cdot -2 \cdot -2 \cdot x^{4+4+4+4+4}\\&=-32x^{20}$

Write the definition of this property and one problem down in your notebook.

Simplify each monomial.

#### Example A

$(6x^3)^2$

Solution: $36x^6$

#### Example B

$(2x^3y^3)^3$

Solution: $8x^9y^9$

#### Example C

$(-3x^2y^2z)^4$

Solution: $81x^8y^8z^4$

Now let's go back to the dilemma from the beginning of the Concept.

Here is the side length of the square platform.

$6a^2$

We want to find the area of the platform. To figure out the area, we will use the following formula.

$A = s^2$

Now we substitute the side length into the formula.

$A = (6a^2)^2$

Next, we can square the monomial.

$36a^4$

### Guided Practice

Here is one for you to try on your own.

$(-2x^4)^5$

Solution

$& (-2x^4)^5 \\&=(-2x^4)(-2x^4)(-2x^4)(-2x^4)(-2x^4) \\&=(-2 \cdot x \cdot x \cdot x \cdot x)(-2 \cdot x \cdot x \cdot x \cdot x)(-2 \cdot x \cdot x \cdot x \cdot x)(-2 \cdot x \cdot x \cdot x \cdot x) \\&=-2 \cdot -2 \cdot -2 \cdot -2 \cdot -2 \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \\&=-32x^{20}$

### Explore More

Directions: Simplify.

1. $(6x^5)^2$
2. $(-13d^5)^2$
3. $(-3p^3 q^4)^3$
4. $(10xy^2)^4$
5. $(-4t^3)^5$
6. $(18 r^2 s^3)^2$
7. $(2r^{11}s^3 t^2)^3$
8. $(7x^2)^2$
9. $(2y^2)^3$
10. $(5x^2)^3$
11. $(12y^3)^2$
12. $(5x^5)^5$
13. $(2x^2y^2z)^3$
14. $(3x^4y^3z^2)^3$
15. $(-5x^4y^3z^3)^3$

### 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.
Coefficient

Coefficient

A coefficient is the number in front of a variable.
Expanded Form

Expanded Form

Expanded form refers to a base and an exponent written as repeated multiplication.
Exponent

Exponent

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

Monomial

A monomial is an expression made up of only one term.
Power

Power

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

Product of Powers Property

The product of powers property states that $a^m \cdot a^n = a^{m+n}$.
Variable

Variable

A variable is a symbol used to represent an unknown or changing quantity. The most common variables are a, b, x, y, m, and n.