Fields Medal Prize Winners (1998)


Solving Quadratic Equations by Using the Quadratic Formula
Addition with Negative Numbers
Solving Linear Systems of Equations by Elimination
Rational Exponents
Solving Quadratic Inequalities
Systems of Equations That Have No Solution or Infinitely Many Solutions
Dividing Polynomials by Monomials and Binomials
Polar Representation of Complex Numbers
Solving Equations with Fractions
Quadratic Expressions Completing Squares
Graphing Linear Inequalities
Square Roots of Negative Complex Numbers
Simplifying Square Roots
The Equation of a Circle
Fractional Exponents
Finding the Least Common Denominator
Simplifying Square Roots That Contain Whole Numbers
Solving Quadratic Equations by Completing the Square
Graphing Exponential Functions
Decimals and Fractions
Adding and Subtracting Fractions
Adding and Subtracting Rational Expressions with Unlike Denominators
Quadratic Equations with Imaginary Solutions
Graphing Solutions of Inequalities
FOIL Multiplying Polynomials
Multiplying and Dividing Monomials
Order and Inequalities
Exponents and Polynomials
Variables and Expressions
Multiplying by 14443
Dividing Rational Expressions
Division Property of Radicals
Equations of a Line - Point-Slope Form
Rationalizing the Denominator
Imaginary Solutions to Equations
Multiplying Polynomials
Multiplying Monomials
Adding Fractions
Rationalizing the Denominator
Rational Expressions
Ratios and Proportions
Rationalizing the Denominator
Like Radical Terms
Adding and Subtracting Rational Expressions With Different Denominators
Percents and Fractions
Reducing Fractions to Lowest Terms
Subtracting Mixed Numbers with Renaming
Simplifying Square Roots That Contain Variables
Factors and Prime Numbers
Rules for Integral Exponents
Multiplying Monomials
Graphing an Inverse Function
Factoring Quadratic Expressions
Solving Quadratic Inequalities
Factoring Polynomials
Multiplying Radicals
Simplifying Fractions 1
Graphing Compound Inequalities
Rationalizing the Denominator
Simplifying Products and Quotients Involving Square Roots
Standard Form of a Line
Multiplication by 572
Adding and Subtracting Fractions
Multiplying Polynomials
Factoring Trinomials
Solving Exponential Equations
Solving Equations with Fractions
Simplifying Complex Fractions
Multiplying and Dividing Fractions
Mathematical Terms
Solving Quadratic Equations by Factoring
Factoring General Polynomials
Adding Rational Expressions with the Same Denominator
The Trigonometric Functions
Solving Nonlinear Equations by Factoring
Solving Systems of Equations
Midpoint of a Line Segment
Complex Numbers
Graphing Systems of Equations
Reducing Rational Expressions
Rewriting Algebraic Fractions
Rationalizing the Denominator
Adding, Subtracting and Multiplying Polynomials
Radical Notation
Solving Radical Equations
Positive Integral Divisors
Solving Rational Equations
Rational Exponents
Mathematical Terms
Rationalizing the Denominator
Subtracting Rational Expressions with the Same Denominator
Axis of Symmetry and Vertex of a Parabola
Simple Partial Fractions
Simplifying Radicals
Powers of Complex Numbers
Fields Medal Prize Winners (1998)

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Rationalizing the Denominator


Examples with Solutions


Example 1:



Since it is a basic property of radicals that

we see that the given expression actually amounts to a fraction with a radical in the denominator. Therefore, to accomplish simplification of this expression, we need to rationalize the denominator:

A quick check before moving on indicates that there is a perfect square factor in the square root that remains, and so we remove this factor from the square root before declaring a final answer:

There really is nothing else we can do to this final expression to simplify it further, so it must be the required final answer.


Example 2:

Write the result of the following division in simplified form:


We deal with division involving radicals in detail in a later document in this series. However, you are already familiar with the way fractions express division, so that here we can write

The only possible simplification at this stage is to rationalize the denominator, which can be accomplished by multiplying the numerator and denominator by :

as the required simplest final result.


Example 3:

Write the result of the following division in simplest form:


This example is very similar to the previous Example 6, and so you should use it as a practice problem. Attempt to work out the final answer yourself before looking at the steps of our solution, which follows.


Now, we rationalize the denominator by multiplying the numerator and denominator by to get

as the required final answer in simplified form.


Example 4:

The formula for the area, A, of a circle of radius r is given by

Derive a formula for the radius, r, in terms of the area, A, and make sure that your result satisfies the usual criteria of simplicity.


We deal systematically with the topic of rearranging formulas in another document on this website. However, we can handle this problem without having to invoke a general strategy. Since

we can write

by just dividing both sides of the original formula by . Now we have a formula for r 2. To get a formula for r, we need to take the square root of both sides:

This formula looks quite simple already, but it does consist of the square root of a fraction, and so the usual rules of simplification say we should rationalize the denominator. This is easily done:

Thus, the final simplified formula for r is