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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Questions & Answers

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what is microbiologist

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what is errata

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is the branch of biology that deals with the study of microorganisms.

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studies of microbes

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Bacteria doesn't produce energy they are dependent upon their substrate in case of lack of nutrients they are able to make spores which helps them to sustain in harsh environments

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But not all bacteria make spores, l mean Eukaryotic cells have Mitochondria which acts as powerhouse for them, since bacteria don't have it, what is the substitution for it?

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they make spores

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the significance of food webs for disease transmission

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food webs brings about an infection as an individual depends on number of diseased foods or carriers dully.

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Assimilatory nitrate reduction is a process that occurs in some microorganisms, such as bacteria and archaea, in which nitrate (NO3-) is reduced to nitrite (NO2-), and then further reduced to ammonia (NH3).

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This process is called assimilatory nitrate reduction because the nitrogen that is produced is incorporated in the cells of microorganisms where it can be used in the synthesis of amino acids and other nitrogen products

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Examples of thermophilic organisms

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Give Examples of thermophilic organisms

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advantages of normal Flora to the host

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Prevent foreign microbes to the host

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they provide healthier benefits to their hosts

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They are friends to host only when Host immune system is strong and become enemies when the host immune system is weakened . very bad relationship!

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cell is the smallest unit of life

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cell is the structural and functional unit of life

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is the fundamental units of Life

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Many sites of the body have it Skin Nasal cavity Oral cavity Gastro intestinal tract

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skin

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all

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part of a tissue or an organ being wounded or bruised.

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Binomial nomenclature

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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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