Exponential Functions
Linera Equations
Simple Trinomials as Products of Binomials
Laws of Exponents and Dividing Monomials
Solving Equations
Multiplying Polynomials
Multiplying and Dividing Rational Expressions
Solving Systems of Linear Inequalities
Mixed-Number Notation
Linear Equations and Inequalities in One Variable
The Quadratic Formula
Fractions and Decimals
Graphing Logarithmic Functions
Multiplication by 111
Solving Systems of Equations - Two Lines
Solving Nonlinear Equations by Factoring
Solving Linear Systems of Equations by Elimination
Rationalizing the Denominator
Simplifying Complex Fractions
Factoring Trinomials
Linear Relations and Functions
Axis of Symmetry and Vertices
Equations Quadratic in Form
The Appearance of a Polynomial Equation
Subtracting Reverses
Non-Linear Equations
Exponents and Order of Operations
Factoring Trinomials by Grouping
Factoring Trinomials of the Type ax 2 + bx + c
The Distance Formula
Invariants Under Rotation
Multiplying and Dividing Monomials
Solving a System of Three Linear Equations by Elimination
Multiplication by 25
Powers of i
Solving Quadratic and Polynomial Equations
Slope-intercept Form for the Equation of a Line
Equations of Lines
Square Roots
Integral Exponents
Product Rule for Radicals
Solving Compound Linear Inequalities
Axis of Symmetry and Vertices
Multiplying Rational Expressions
Reducing Rational Expressions
Properties of Negative Exponents
Numbers, Factors, and Reducing Fractions to Lowest Terms
Solving Quadratic Equations
Factoring Completely General Quadratic Trinomials
Solving a Formula for a Given Variable
Factoring Polynomials
Decimal Numbers and Fractions
Multiplication Properties of Exponents
Multiplying Fractions
Multiplication by 50

Try the Free Math Solver or Scroll down to Tutorials!












Please use this form if you would like
to have this math solver on your website,
free of charge.

What happens when we divide powers? Let's analyze this by dividing various powers of 2.

Try to make a conjecture about dividing powers that have the same base.

In both cases, the result is the original base raised to the power given by the difference of the two exponents.

For ,

and for .

The table on the left shows what happens when a power of 2 is divided by 2 1 . The table on the right shows what happens when a power of 2 is divided by 2 2

In the table on the left, notice the powers that result. Each resulting power can be found by subtracting 1 from the exponent of the original power. In the table on the right, each resulting power can be found by subtracting 2 from the exponent of the original power. This agrees with our original observation that when we divide two powers with the same base, the exponent of the resulting power is the difference of the exponents of the two dividends.

Consider another case. Let's divide a 4 by a 2. To do this, expand a 4 into a · a · a · a and a 2 into a · a . Next place them into the fraction

We can now cancel two a's from both numerator and denominator.

Cancellation is a shorthand process involving the properties of fractions. Also, point out that any number raised to the first power is that number itself.

After canceling, we find that .

In general, when we write the quotient and expand it into products of a's, the result is

which shows why this fact is valid.

Key Idea

When we divide a power of a by another smaller power of a , the result is a power of a , in which the exponent is the difference of the exponents of the two dividends. In symbols,

This holds true for any nonzero number a and whole numbers b and c.