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By the end of this section, you will be able to:
  • Simplify expressions with exponents
  • Simplify expressions using the Product Property of Exponents
  • Simplify expressions using the Power Property of Exponents
  • Simplify expressions using the Product to a Power Property
  • Simplify expressions by applying several properties
  • Multiply monomials

Before you get started, take this readiness quiz.

  1. Simplify: 3 4 · 3 4 .
    If you missed the problem, review Multiply and Divide Fractions .
  2. Simplify: ( −2 ) ( −2 ) ( −2 ) .
    If you missed the problem, review Multiply and Divide Integers .

Simplify expressions with exponents

Remember that an exponent indicates repeated multiplication of the same quantity. For example, 2 4 means to multiply four factors of 2 , so 2 4 means 2 · 2 · 2 · 2 . This format is known as exponential notation .

Exponential notation

On the left side, a raised to the m is shown. The m is labeled in blue as an exponent. The a is labeled in red as the base. On the right, it says a to the m means multiply m factors of a. Below this, it says a to the m equals a times a times a times a, with m factors written below in blue.

This is read a to the m th power.

In the expression a m , the exponent tells us how many times we use the base a as a factor.

On the left side, 7 to the 3rd power is shown. Below is 7 times 7 times 7, with 3 factors written below. On the right side, parentheses negative 8 to the 5th power is shown. Below is negative 8 times negative 8 times negative 8 times negative 8 times negative 8, with 5 factors written below.

Before we begin working with variable expressions containing exponents, let’s simplify a few expressions involving only numbers.

Simplify:

  1. 5 3
  2. 9 1

Solution

5 3
Multiply 3 factors of 5. 5 · 5 · 5
Simplify. 125
9 1
Multiply 1 factor of 9. 9
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Simplify:

  1. 4 3
  2. 11 1

  1. 64
  2. 11

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Simplify:

  1. 3 4
  2. 21 1

  1. 81
  2. 21

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Simplify:

  1. ( 7 8 ) 2
  2. ( 0.74 ) 2

Solution

( 7 8 ) 2
Multiply two factors. ( 7 8 ) ( 7 8 )
Simplify. 49 64
( 0.74 ) 2
Multiply two factors. ( 0.74 ) ( 0.74 )
Simplify. 0.5476
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Simplify:

  1. ( 5 8 ) 2
  2. ( 0.67 ) 2

  1. 25 64
  2. 0.4489

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Simplify:

  1. ( 2 5 ) 3
  2. ( 0.127 ) 2

  1. 8 125
  2. 0.016129

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Simplify:

  1. ( −3 ) 4
  2. −3 4

Solution

( −3 ) 3
Multiply four factors of −3. ( −3 ) ( −3 ) ( −3 ) ( −3 )
Simplify. 81
−3 4
Multiply two factors. ( 3 · 3 · 3 · 3 )
Simplify. −81

Notice the similarities and differences in parts and . Why are the answers different? In part the parentheses tell us to raise the (−3) to the 4 th power. In part we raise only the 3 to the 4 th power and then find the opposite.

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Simplify:

  1. ( −2 ) 4
  2. −2 4

  1. 16
  2. −16

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Simplify:

  1. ( −8 ) 2
  2. −8 2

  1. 64
  2. −64

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Simplify expressions using the product property of exponents

You have seen that when you combine like terms by adding and subtracting, you need to have the same base with the same exponent. But when you multiply and divide, the exponents may be different, and sometimes the bases may be different, too. We’ll derive the properties of exponents by looking for patterns in several examples. All the exponent properties hold true for any real numbers, but right now we will only use whole number exponents.

First, we will look at an example that leads to the Product Property.

.
What does this mean?

How many factors altogether?
.
So, we have .
Notice that 5 is the sum of the exponents, 2 and 3. .

The base stayed the same and we added the exponents. This leads to the Product Property for Exponents.

Product property of exponents

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

a m · a n = a m + n

To multiply with like bases, add the exponents.

An example with numbers helps to verify this property.

2 2 · 2 3 = ? 2 2 + 3 4 · 8 = ? 2 5 32 = 32

Simplify: x 5 · x 7 .

Solution

x 5 · x 7
Use the product property, a m · a n = a m + n . .
Simplify. x 12
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Simplify: x 7 · x 8 .

x 15

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Simplify: x 5 · x 11 .

x 16

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Simplify: b 4 · b .

Solution

b 4 · b
Rewrite, b = b 1 . b 4 · b 1
Use the product property, a m · a n = a m + n . .
Simplify. b 5
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Simplify: p 9 · p .

p 10

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Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
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Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
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da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
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I only see partial conversation and what's the question here!
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what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
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LITNING
scanning tunneling microscope
Sahil
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Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
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nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
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why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
Kimberly Reply
Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
August Reply
What is the expressiin for seven less than four times the number of nickels
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how do you translate this in Algebraic Expressions
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Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
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. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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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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