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Exponential Decay
Negative Exponents
Multiplying and Dividing Fractions 4
Evaluating Expressions Involving Fractions
The Cartesian Coordinate System
Adding and Subtracting Fractions with Like Denominators
Solving Absolute Value Inequalities
Multiplying Special Polynomials
FOIL Method
Inequalities
Solving Systems of Equations by Graphing
Graphing Compound Inequalities
Solving Quadratic Equations by Completing the Square
Addition Property of Equality
Square Roots
Adding and Subtracting Fractions
The Distance Formula
Graphing Logarithmic Functions
Fractions
Dividing Mixed Numbers
Evaluating Polynomials
Power of a Product Property of Exponents
Terminology of Algebraic Expressions
Adding and Subtracting Rational Expressions with Identical Denominators
Solving Exponential Equations
Factoring The Difference of 2 Squares
Changing Fractions to Decimals
Solving Linear Equations
Using Patterns to Multiply Two Binomials
Completing the Square
Roots of Complex Numbers
Methods for Solving Quadratic Equations
Conics in Standard Form
Solving Quadratic Equations by Using the Quadratic Formula
Simplifying Fractions 2
Exponential Notation
Exponential Growth
The Cartesian Plane
Graphing Linear Functions
The Slope of a Line
Finding Cube Roots of Large Numbers
Rotating Axes
Common Mistakes With Percents
Solving an Equation That Contains a Square Root
Rational Equations
Properties of Common Logs
Composition of Functions
Using Percent Equations
Solving Inequalities
Properties of Exponents
Graphing Quadratic Functions
Factoring a Polynomial by Finding the GCF
The Rectangular Coordinate System
Adding and Subtracting Fractions
Multiplying and Dividing Rational Expressions
Improper Fractions and Mixed Numbers
Properties of Exponents
Complex Solutions of Quadratic Equations
Solving Nonlinear Equations by Factoring
Solving Quadratic Equations by Factoring
Least Common Multiples
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Solving Exponential Equations
Solving Linear Equations
Multiplication Property of Equality
Multiplying Mixed Numbers
Multiplying Fractions
Reducing a Rational Expression to Lowest Terms
Literal Numbers
Factoring Trinomials
Logarithmic Functions
Adding Fractions with Unlike Denominators
Simplifying Square Roots
Adding Fractions
Equations Quadratic in Form
Dividing Rational Expressions
Slopes of Parallel Lines
Simplifying Cube Roots That Contain Variables
Functions and Graphs
Complex Numbers
Multiplying and Dividing Fractions 1
Composition of Functions
Intercepts of a Line
Powers
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Multiplying Two Numbers with the same Tens Digit and whose Ones Digits add up to 10
Factoring Trinomials
Exponents and Polynomials
Decimals and their Equivalent Fractions
Negative Integer Exponents
Adding and Subtracting Mixed Numbers
Solving Quadratic Equations
Theorem of Pythagoras
Equations 1
Subtracting Fractions
Solving Quadratic Equations by Graphing
Evaluating Polynomials
Slope
Angles and Degree Measure
   
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Negative Exponents

We started out by defining powers to be repeated multiplications:

From this definition of the power or exponential notation, several laws or properties could be inferred for powers of products and quotients of numbers. These were described in the preceding document in these notes.

One of those laws dealt with the quotient of two powers with the same base. For example

Of course, if the bigger power is in the denominator, the result has the net power in the denominator:

So far, both of these make sense when we consider powers to be repeated multiplications, since only positive exponents occur in both cases. However, if we simply subtracted the denominator exponent from the numerator exponent in the second example, we would have got

(*)

At first, this result, with the negative exponent, appears a bit nonsensical. We know how to multiply 2 by itself +3 times, but it’s hard to imagine what it might mean to multiply 2 by itself ‘-3’ times. The most useful way to get around this apparent nonsense is adopt the rule that whenever we write 2 -3, we really mean , that is:

With this convention, results such as the one labelled (*) make perfect sense.

So, if c is a nonzero number and n is a positive number (so that –n is a negative number), we define

Remarks:

(i) This definition is consistent with all of the laws of exponents given earlier, so the laws may be used with positive and negative exponents.

(ii) When powers occur as a factor in the numerator or the denominator of a fraction, the factors can be switched from top to bottom or vice versa by changing the sign of the exponent. Thus

(iii) Now we see, for example, that

since the numerator and denominator are identical. By comparing these last two expressions, we see that we can make sense of 2 0 by defining it to be equal to 1, ie.

2 0 = 1

In general, if c is a nonzero number, we will define

With this, we now have a reasonable way to interpret powers in which the exponent is any whole number, positive or negative, or zero. This will make the laws of exponents very helpful in simplifying complicated expressions involving powers.

 

example:

or

 

example:

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