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# Exponential Properties Involving Products

## Add exponents to multiply exponents by other exponents

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Recognize and Apply the Power of a Product Property

Simone is building a platform for the stage of the new band stand. She needs to determine the area of the platform so she can order the wood she needs. She knows the platform which has a side length of \begin{align*}6a^2\end{align*} will be square.

How can she find the area of the platform?

In this concept, you will learn to recognize and apply the power of a product property.

### Power of Product Property

When multiplying monomials, an exponent is applied to the constant, variable, or quantity that is directly to its left.

Let’s look at an example where the exponents can be applied to products using parentheses.

\begin{align*}(5x)^4\end{align*}

If you apply the exponent 4 to whatever is directly to its left, apply it to the parentheses, not just to the \begin{align*}x\end{align*}.

The parentheses are directly to the left of the 4. This indicates that the entire product in the parentheses is taken to the 4th power.

First, write \begin{align*}(5x)^4\end{align*} in expanded form.

\begin{align*}(5x)^4 = (5x)(5x)(5x)(5x)\end{align*}

Next, multiply the monomials by placing like factors next to each other, multiplying the coefficients, and simplifying using exponents.

\begin{align*}\begin{array}{rcl} (5x)^4 &=& (5x)(5x)(5x)(5x) \\ &=& 5 \cdot 5 \cdot 5 \cdot 5 \cdot x \cdot x \cdot x \cdot x \\ &=& 625x^4 \end{array}\end{align*}

This is the Power of a Product Property which says, for any nonzero numbers \begin{align*}a\end{align*} and \begin{align*}b\end{align*} and any integer \begin{align*}n\end{align*}:

\begin{align*}(ab)^n = a^n b^n\end{align*}

Let’s look at an example.

Use the Power of a Product Property to expand \begin{align*}(7h)^3\end{align*}.

First, expand the parentheses by multiplying \begin{align*}7h\end{align*} times itself, three times.

\begin{align*}(7h)^3 = (7h)(7h)(7h)\end{align*}

Next, multiply the monomials by placing like factors next to each other, multiplying the coefficients, and simplifying using exponents.

\begin{align*}\begin{array}{rcl} (7h)^3 &=& (7h)(7h)(7h) \\ &=& 7 \cdot 7 \cdot 7 \cdot h \cdot h \cdot h \\ &=& 343 h^3 \end{array}\end{align*}

The answer is \begin{align*}343 h^3\end{align*}.

There is a definite pattern between the exponents and the final product. When you multiply like bases, there is a shortcut-add the exponents of like bases. Another way of saying it is:

\begin{align*}a^m \times a^n = a^{m+n}\end{align*}

Let’s look at another problem.

Use the Power of a Product Property to expand \begin{align*}(-2x^4)^5\end{align*}.

First, expand the parentheses by multiplying the base of \begin{align*}-2x^4\end{align*} by itself, five times.

\begin{align*}(-2x^4)^5 = (-2x^4)(-2x^4)(-2x^4)(-2x^4)(-2x^4)\end{align*}

Next, multiply the monomials by placing like factors next to each other, multiplying the coefficients, and simplifying using exponents.

\begin{align*}\begin{array}{rcl} (-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} \\ &=& -32 x^{20} \end{array}\end{align*}

The answer is \begin{align*}(-32x)^{20}\end{align*}.

### Examples

#### Example 1

Earlier, you were given a problem about Simone and the square platform. She needs to figure out the area of the platform to order the needed wood.

The side length of the square platform is \begin{align*}6a^2\end{align*}.

First, set up the area of the square platform.

\begin{align*}\begin{array}{rcl} A &=& s^2 \\ A &=& (6a^2)^2 \end{array}\end{align*}

Next, expand the parentheses by multiplying the monomial times itself two times.

\begin{align*}(6a^2)^2 = (6a^2)(6a^2)\end{align*}

Then, multiply the monomials by placing like factors next to each other, multiplying the coefficients, and simplifying using exponents.

\begin{align*}\begin{array}{rcl} (6a^2)^2 &=& (6a^2)(6a^2) \\ &=& 6 \cdot 6 \cdot a^2 \cdot a^2 \\ &=& 6 \cdot 6 \cdot a^{2+2} \\ &=& 36a^4 \end{array}\end{align*}

The answer is \begin{align*}36a^4\end{align*}.

The area of the square platform is \begin{align*}36a^4\end{align*} units squared.

#### Example 2

Use the Power of a Product Property to expand \begin{align*}(3x^5)^3\end{align*}.

First, expand the parentheses by multiplying \begin{align*}3x^5\end{align*} times itself, three times.

\begin{align*}(3x^5)^3 = (3x^5)(3x^5)(3x^5)\end{align*}

Next, multiply the monomials by placing like factors next to each other, multiplying the coefficients, and simplifying using exponents.

\begin{align*}\begin{array}{rcl} (3x^5)^3 &=& (3x^5)(3x^5)(3x^5) \\ &=& 3 \cdot 3 \cdot 3 \cdot x^5 \cdot x^5 \cdot x^5 \\ &=& 3 \cdot 3 \cdot 3 \cdot x^{5+5+5} \\ &=& 27x^{15} \end{array}\end{align*}

The answer is \begin{align*}27x^{15}\end{align*}.

#### Example 3

Simplify the monomial \begin{align*}(6x^3)^2\end{align*}.

First, expand the parentheses by multiplying \begin{align*}6x^3\end{align*} times itself, two times.

\begin{align*}(6x^3)^2 = (6x^3)(6x^3)\end{align*}

Next, multiply the monomials by placing like factors next to each other, multiplying the coefficients, and simplifying using exponents.

\begin{align*}\begin{array}{rcl} (6x^3)^2 &=& (6x^3)(6x^3) \\ &=& 6 \cdot 6 \cdot x^3 \cdot x^3 \\ &=& 6 \cdot 6 \cdot x^{3+3} \\ &=& 36 x^6 \end{array}\end{align*}

The answer is \begin{align*}36x^6\end{align*}.

#### Example 4

Simplify the monomial \begin{align*}(2x^3 y^3)^3\end{align*}.

First, expand the parentheses by multiplying the monomial times itself, three times.

\begin{align*}(2x^3 y^3)^3 = (2x^3 y^3)(2x^3 y^3)(2x^3 y^3)\end{align*}

Next, multiply the monomials by placing like factors next to each other, multiplying the coefficients, and simplifying using exponents.

\begin{align*}\begin{array}{rcl} (2x^3 y^3)^3 &=& (2x^3 y^3)(2x^3 y^3)(2x^3 y^3) \\ &=& 2 \cdot 2 \cdot 2 \cdot x^3 \cdot x^3 \cdot x^3 \cdot y^3 \cdot y^3 \cdot y^3 \\ &=& 2 \cdot 2 \cdot 2 \cdot x^{3+3+3} \cdot y^{3+3+3} \\ &=& 8x^9 y^9 \end{array} \end{align*}

The answer is \begin{align*}8x^9y^9\end{align*}.

#### Example 5

Simplify the monomial \begin{align*}(-3x^2 y^2 z)^4\end{align*}.

First, expand the parentheses by multiplying the monomial times itself, four times.

\begin{align*}(-3x^2 y^2 z)^4 = (-3x^2 y^2 z)(-3x^2 y^2 z)(-3x^2 y^2 z)(-3x^2 y^2 z)\end{align*}

Next, multiply the monomials by placing like factors next to each other, multiplying the coefficients, and simplifying using exponents.

\begin{align*}\begin{array}{rcl} (-3x^2 y^2 z)^4 &=& (-3x^2 y^2 z)(-3x^2 y^2 z)(-3x^2 y^2 z)(-3x^2 y^2 z) \\ &=& -3 \cdot -3 \cdot -3 \cdot -3 \cdot x^2 \cdot x^2 \cdot x^2 \cdot x^2 \cdot y^2 \cdot y^2 \cdot y^2 \cdot y^2 \cdot z \cdot z \cdot z \cdot z \\ &=& -3 \cdot -3 \cdot -3 \cdot -3 \cdot x^{2+2+2+2} \cdot y^{2+2+2+2} \cdot z^{1+1+1+1} \\ &=& 81 x^8 y^8 z^4 \end{array}\end{align*}

The answer is \begin{align*}81 x^8 y^8 z^4\end{align*}.

### Review

Simplify.

1. \begin{align*}(6x^5)^2\end{align*}
2. \begin{align*}(-13 d^5)^2\end{align*}
3. \begin{align*}(-3 p^3 q^4)^3\end{align*}
4. \begin{align*}(1 0 xy^2)^4\end{align*}
5. \begin{align*}(-4t^3)^5\end{align*}
6. \begin{align*}(18 r^2 s^3)^2\end{align*}
7. \begin{align*}(2r^{11} s^3 t^2)^3\end{align*}
8. \begin{align*}(7x^2)^2\end{align*}
9. \begin{align*}(2y^2)^3\end{align*}
10. \begin{align*}(5x^2)^3\end{align*}
11. \begin{align*}(12 y^3)^2\end{align*}
12. \begin{align*}(5x^5)^5\end{align*}
13. \begin{align*}(2x^2 y^2 z)^3\end{align*}
14. \begin{align*}(3x^4 y^3 z^2)^3\end{align*}
15. \begin{align*}(-5x^4 y^3 z^3)^3\end{align*}

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### Vocabulary Language: English

TermDefinition
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 A coefficient is the number in front of a variable.
Expanded Form Expanded form refers to a base and an exponent written as repeated multiplication.
Exponent Exponents are used to describe the number of times that a term is multiplied by itself.
Monomial A monomial is an expression made up of only one term.
Power The "power" refers to the value of the exponent. For example, $3^4$ is "three to the fourth power".
Product of Powers Property The product of powers property states that $a^m \cdot a^n = a^{m+n}$.
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.