10.2 Use multiplication properties of exponents (Page 2/3)

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Simplify: $m \cdot m^{7} .$

m ⁸

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Simplify: $2^{7} \cdot 2^{9} .$

Solution

	$2^{7} \cdot 2^{9}$
Use the product property, $a^{m} \cdot a^{n} = a^{m + n} .$
Simplify.	$2^{16}$

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Simplify: $6 \cdot 6^{9} .$

6 ¹⁰

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Simplify: $9^{6} \cdot 9^{9} .$

9 ¹⁵

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Simplify: $y^{17} \cdot y^{23} .$

Solution

	$y^{17} \cdot y^{23}$
Notice, the bases are the same, so add the exponents.
Simplify.	$y^{40}$

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Simplify: $y^{24} \cdot y^{19} .$

y ⁴³

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Simplify: $z^{15} \cdot z^{24} .$

z ³⁹

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We can extend the Product Property of Exponents to more than two factors.

Simplify: $x^{3} \cdot x^{4} \cdot x^{2} .$

Solution

	$x^{3} \cdot x^{4} \cdot x^{2}$
Add the exponents, since the bases are the same.
Simplify.	$x^{9}$

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Simplify: $x^{7} \cdot x^{5} \cdot x^{9} .$

x ²¹

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Simplify: $y^{3} \cdot y^{8} \cdot y^{4} .$

y ¹⁵

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Simplify expressions using the power property of exponents

Now let’s look at an exponential expression that contains a power raised to a power. See if you can discover a general property.



What does this mean? How many factors altogether?
So, we have
Notice that 6 is the product of the exponents, 2 and 3.

We multiplied the exponents. This leads to the Power Property for Exponents.

Power property of exponents

If $a$ is a real number and $m, n$ are whole numbers, then

{(a^{m})}^{n} = a^{m \cdot n}

To raise a power to a power, multiply the exponents.

An example with numbers helps to verify this property.

\begin{matrix} {(5^{2})}^{3} & \overset{?}{=} & 5^{2 \cdot 3} \\ {(25)}^{3} & \overset{?}{=} & 5^{6} \\ 15,625 & = & 15,625 ✓ \end{matrix}

Simplify:

ⓐ ${(x^{5})}^{7}$
ⓐ ${(3^{6})}^{8}$

Solution

ⓐ
	${(x^{5})}^{7}$
Use the Power Property, ${(a^{m})}^{n} = a^{m \cdot n} .$
Simplify.	$x^{35}$

ⓑ
	${(3^{6})}^{8}$
Use the Power Property, ${(a^{m})}^{n} = a^{m \cdot n} .$
Simplify.	$3^{48}$

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Simplify:

ⓐ ${(x^{7})}^{4}$
ⓑ ${(7^{4})}^{8}$

ⓐ x ²⁸
ⓑ 7 ³²

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Simplify:

ⓐ ${(x^{6})}^{9}$
ⓑ ${(8^{6})}^{7}$

ⓐ y ⁵⁴
ⓑ 8 ⁴²

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Simplify expressions using the product to a power property

We will now look at an expression containing a product that is raised to a power. Look for a pattern.

	${(2 x)}^{3}$
What does this mean?	$2 x \cdot 2 x \cdot 2 x$
We group the like factors together.	$2 \cdot 2 \cdot 2 \cdot x \cdot x \cdot x$
How many factors of 2 and of $x ?$	$2^{3} \cdot x^{3}$
Notice that each factor was raised to the power.	${(2 x)}^{3} is 2^{3} \cdot x^{3}$
We write:	${(2 x)}^{3}$ $2^{3} \cdot x^{3}$

The exponent applies to each of the factors. This leads to the Product to a Power Property for Exponents.

Product to a power property of exponents

If $a$ and $b$ are real numbers and $m$ is a whole number, then

{(a b)}^{m} = a^{m} b^{m}

To raise a product to a power, raise each factor to that power.

An example with numbers helps to verify this property:

\begin{matrix} {(2 \cdot 3)}^{2} & \overset{?}{=} & 2^{2} \cdot 3^{2} \\ 6^{2} & \overset{?}{=} & 4 \cdot 9 \\ 36 & = & 36 ✓ \end{matrix}

Simplify: ${(−11 x)}^{2} .$

Solution

	${(−11 x)}^{2}$
Use the Power of a Product Property, ${(a b)}^{m} = a^{m} b^{m} .$
Simplify.	$121 x^{2}$

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Simplify: ${(−14 x)}^{2} .$

196 x ²

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Simplify: ${(−12 a)}^{2} .$

144 a ²

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Simplify: ${(3 x y)}^{3} .$

Solution

	${(3 x y)}^{3}$
Raise each factor to the third power.
Simplify.	$27 x^{3} y^{3}$

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Simplify: ${(−4 x y)}^{4} .$

256 x ⁴ y ⁴

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Simplify: ${(6 x y)}^{3} .$

216 x ³ y ³

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Simplify expressions by applying several properties

We now have three properties for multiplying expressions with exponents. Let’s summarize them and then we’ll do some examples that use more than one of the properties.

Properties of exponents

If $a, b$ are real numbers and $m, n$ are whole numbers, then

\begin{matrix} Product Property & a^{m} \cdot a^{n} = a^{m + n} \\ Power Property & {(a^{m})}^{n} = a^{m \cdot n} \\ Product to a Power Property & {(a b)}^{m} = a^{m} b^{m} \end{matrix}

Simplify: ${(x^{2})}^{6} {(x^{5})}^{4} .$

Solution

	${(x^{2})}^{6} {(x^{5})}^{4}$
Use the Power Property.	$x^{12} \cdot x^{20}$
Add the exponents.	$x^{32}$

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Simplify: ${(x^{4})}^{3} {(x^{7})}^{4} .$

x ⁴⁰

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Simplify: ${(y^{9})}^{2} {(y^{8})}^{3} .$

y ⁴²

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Simplify: ${(- 7 x^{3} y^{4})}^{2} .$

Solution

	${(- 7 x^{3} y^{4})}^{2}$
Take each factor to the second power.	${(−7)}^{2} {(x^{3})}^{2} {(y^{4})}^{2}$
Use the Power Property.	$49 x^{6} y^{8}$

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Simplify: ${(- 8 x^{4} y^{7})}^{3} .$

−512 x ¹² y ²¹

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Simplify: ${(- 3 a^{5} b^{6})}^{4} .$

81 a ²⁰ b ²⁴

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Simplify: ${(6 n)}^{2} (4 n^{3}) .$

Solution

	${(6 n)}^{2} (4 n^{3})$
Raise $6 n$ to the second power.	$6^{2} n^{2} \cdot 4 n^{3}$
Simplify.	$36 n^{2} \cdot 4 n^{3}$
Use the Commutative Property.	$36 \cdot 4 \cdot n^{2} \cdot n^{3}$
Multiply the constants and add the exponents.	$144 n^{5}$

Notice that in the first monomial, the exponent was outside the parentheses and it applied to both factors inside. In the second monomial, the exponent was inside the parentheses and so it only applied to the n .

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Source: OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9

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