diverging lens equation Rules of Exponents – The Zero Exponent Rule and the Negative … – HCC

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Exponents and Scientific Notation

Section 1: Product Rule, Quotient Rule, and Power Rules Section 2: Zero Exponents and Negative Exponents Section 3: Scientific Notation Section 4: Rational Exponents Dr. Bump’s Videos on Exponents and Scientific Notation

Math Review

Exponents and Scientific Notation

Section 2: Rules of Exponents – The Zero Exponent Rule and the Negative Exponent Rule

It Is Important That You Watch This Video First

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Negative exponents and zero exponents often show up when applying formulas or simplifying expressions.

In this section, we will define the Negative Exponent Rule and the Zero Exponent Rule and look at a couple of examples.


Negative Exponent Rule:

In other words, when there is a negative exponent, we need to create a fraction and put the exponential expression in the denominator and make the exponent positive. For example,

But working with negative exponents is just rule of exponents that we need to be able to use when working with exponential expressions.

  Example:

Simplify: 3-2

Solution:

3-2 =


  Example:

Simplify:

Solution:

Apply the Negative Exponent Rule to both the numerator and the denominator.


  Example:

Simplify: 3-1 + 5-1

Solution:

Apply the Negative Exponent Rule to each term and then add fractions by finding common denominators.


Zero Exponent Rule: a0 = 1, a not equal to 0. The expression 00 is indeterminate, or undefined.

In the following example, when we apply the product rule for exponents, we end up with an exponent of zero.

x5x-5 = x5 + (-5) = x0

To help understand the purpose of the zero exponent, we will also rewrite x5x-5 using the negative exponent rule.

x5x-5 =

The zero exponent indicates that there are no factors of a number.


  Example:

Simplify each of the following expressions using the zero exponent rule for exponents. Write each expression using only positive exponents.

a) 30

b) -30 + n0

Solution:

a) Apply the Zero Exponent Rule.

30 = 1

b) Apply the Zero Exponent Rule to each term, and then simplify. The zero exponent on the first term applies to the 3 only and not the negative in front of the 3.

-30 + n0 =-(30) + n0 = – 1 + 1 = 0


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A negative exponent indicates that the reciprocal of the base is needed in
order to rewrite the expression with a positive exponent. In the following
examples, the expressions will contain only factors. Therefore when you take
the reciprocal of the base, you can simply move that factor from the numerator
to the denominator and vise versa.

With addition, taking the reciprocal of the base must stay in the same location.

Another Exponent Rule:

a-m
=    1   


            am

If the base is raised to a negative exponent,
take the reciprocal of
the base.
Click here for the first example.
Look at the expression
and locate the negative exponent.

2a-3b

What is the negative exponent?

Click here .

 

 

 

 

 

 

 

 

 

Look at
the expression and locate the negative exponent.

2a-3b

What is the negative exponent? -3

Click here .
Look at
the expression and locate the negative exponent.

2a-3b

What is the negative exponent? -3

What is the base?

Click here .
Look at
the expression and locate the negative exponent.

2a-3b

What is the negative exponent? -3

What is the base? a

Click here .

 

 

 

 

 

 

 

Look at
the expression and locate the negative exponent.

2a-3b

What is the negative exponent? -3

What is the base? a

 

To rewrite the negative exponent as a positive exponent,
take the reciprocal of the base a.

Click here .

 

 

 

 

 

 

 

Look at
the expression and locate the negative exponent.

2a-3b

 

 

2b
a3

What is the negative exponent? -3

What is the base? a

 

To rewrite the negative exponent as a positive exponent,
take the reciprocal of the base a.

Please make a note that the exponent is now positive three.

Click here .

 

 

 

 

 

 

 

A negative exponent
can be rewritten as a positive exponent if you take the reciprocal of the
base.

(a-2b3)-4

First rewrite outside exponents as positive exponents.

Click here .

 

 

 

 

 

 

A negative exponent
can be rewritten as a positive exponent if you take the reciprocal of the
base.

(a-2b3)-4

    1    
(a-2b3)4

First rewrite outside exponents as positive exponents.

Click here .

 

 

 

 

 

 

 

A negative exponent
can be rewritten as a positive exponent if you take the reciprocal of the
base.

(a-2b3)-4

    1    
(a-2b3)4

First rewrite outside exponents as positive exponents.

Did you notice that the base was everything
inside the parenthesis?

Click here .

 

 

 

 

 

 

 

A negative exponent
can be rewritten as a positive exponent if you take the reciprocal of the
base.

(a-2b3)-4

    1    
(a-2b3)4

First rewrite outside exponents as positive exponents.

Did you notice that the base was everything inside the parenthesis?

Clear the outside exponent by multiplying the base to itself four times.(a-2b3)(a-2b3)(a-2b3)(a-2b3)

Click here .

 

 

 

 

 

 

 

A negative exponent
can be rewritten as a positive exponent if you take the reciprocal of the
base.

(a-2b3)-4

    1    
(a-2b3)4

    1    
a-8b12

First rewrite outside exponents as positive exponents.

Did you notice that the base was everything inside the parenthesis?

Clear the outside exponent by multiplying the base to itself four times.(a-2b3)(a-2b3)(a-2b3)(a-2b3)

Click here .

 

 

 

 

 

 

 

A negative exponent
can be rewritten as a positive exponent if you take the reciprocal of the
base.

(a-2b3)-4

    1    
(a-2b3)4

    1    
a-8b12

First rewrite outside exponents as positive exponents.

Did you notice that the base was everything inside the parenthesis?

Clear the outside exponent by multiplying the base to itself four times.(a-2b3)(a-2b3)(a-2b3)(a-2b3)

Are there any negative exponents left in the expression?

Click here .

 

 

 

 

 

 

 

A negative exponent
can be rewritten as a positive exponent if you take the reciprocal of the
base.

(a-2b3)-4

    1    
(a-2b3)4

    1    
a-8b12

First rewrite outside exponents as positive exponents.

Did you notice that the base was everything inside the parenthesis?

Clear the outside exponent by multiplying the base to itself four times.(a-2b3)(a-2b3)(a-2b3)(a-2b3)

Are there any negative exponents left in the expression?
yes

Click here .

 

 

 

 

 

 

 

A negative exponent
can be rewritten as a positive exponent if you take the reciprocal of the
base.

(a-2b3)-4

    1    
(a-2b3)4

    1    
a-8b12

First rewrite outside exponents as positive exponents.

Did you notice that the base was everything inside the parenthesis?

Clear the outside exponent by multiplying the base to itself four times.(a-2b3)(a-2b3)(a-2b3)(a-2b3)

Are there any negative exponents left in the expression?
yes

What is the base of the negative exponent?

Click here .

 

 

 

 

 

 

 

A negative exponent
can be rewritten as a positive exponent if you take the reciprocal of the
base.

(a-2b3)-4

    1    
(a-2b3)4

    1    
a-8b12

First rewrite outside exponents as positive exponents.

Did you notice that the base was everything inside the parenthesis?

Clear the outside exponent by multiplying the base to itself four times.(a-2b3)(a-2b3)(a-2b3)(a-2b3)

Are there any negative exponents left in the expression? yes

What is the base of the negative exponent? a

Click here .

 

 

 

 

 

 

A negative exponent
can be rewritten as a positive exponent if you take the reciprocal of the
base.

(a-2b3)-4

    1    
(a-2b3)4

    1    
a-8b12

First rewrite outside exponents as positive exponents.

Did you notice that the base was everything inside the parenthesis?

Clear the outside exponent by multiplying the base to itself four times.(a-2b3)(a-2b3)(a-2b3)(a-2b3)

Are there any negative exponents left in the expression? yes

What is the base of the negative exponent? a

What does the final result look like?

Click here .

 

 

 

 

 

 

 

 
A negative exponent
can be rewritten as a positive exponent if you take the reciprocal of the
base.

(a-2b3)-4

    1    
(a-2b3)4

    1    
a-8b12

First rewrite outside exponents as positive exponents.

Did you notice that the base was everything inside the parenthesis?

Clear the outside exponent by multiplying the base to itself four times.(a-2b3)(a-2b3)(a-2b3)(a-2b3)

Are there any negative exponents left in the expression? yes

What is the base of the negative exponent? a

What does the final result look like?

a to the 8 divided by b to the 12
Click here
for another example.