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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

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Adin Reply
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Kyle
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Adin
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Adin
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biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
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Introduction about quantum dots in nanotechnology
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nano basically means 10^(-9). nanometer is a unit to measure length.
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there is no specific books for beginners but there is book called principle of nanotechnology
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Devang Reply
are you nano engineer ?
s.
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Tarell
what is the actual application of fullerenes nowadays?
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That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
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Mostly, they use nano carbon for electronics and for materials to be strengthened.
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carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
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of graphene you mean?
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or in general
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in general
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Graphene has a hexagonal structure
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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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