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Fields Medal Prize Winners (1998)




TUTORIALS:


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
Fractions
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
Roots
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
Powers
Rewriting Algebraic Fractions
Exponents
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:

Simplify

solution:

There is no factorization of the expression in the square root possible here, so the only simplification necessary is to rationalize the denominator of the fraction. Since the square root in the denominator is already as simple as possible, we need to just multiply the numerator and denominator by that square root to achieve the required result.

Although there is a t in both the numerator and the denominator, it is not a factor in the denominator, and therefore we cannot cancel the t’s between the numerator and denominator of this last form. In fact, there is obviously no common factors to cancel between the numerator and denominator of this last form, so this last expression is as simple as possible and so it must be the required final answer.

NOTE: Sometimes as they focus on simplifying fractions such as the final result of Example 1 above, people note that the numerator and denominator contain a power of the same expression – in this case of (s + t). Without thinking they do the following:

or

Superficially, it may appear that one or both of these lines represents a useful simplification, but in fact, both lines really just amount to reversing the simplification previously done. The second line obviously just gives back the original problem, and the first line does as well, though in a somewhat less recognizable form. So, while it is very useful to break the simplification of complicated algebraic expressions down into a series of relatively simple independent steps, it is also important to review your entire solution to make sure that you haven’t lost track of what the problem actually requires you to accomplish, and make sure that that is indeed what your final answer has accomplished.

 

Example 2:

Rationalize the denominator in and simplify the result as much as possible.

solution:

If we simply follow the steps illustrated in the first and second examples above, we get

which has a rational denominator, but clearly can be simplified further. First

8x 3 = 2 2 · 2 · x 2 · x

so

Thus,

as the final simplified result.

Alternatively, we could have noted that the square root in the denominator of the original fraction is not in simplest form, and so we could have started by simplifying both the square root and the fraction itself first:

Now the denominator of this fraction can be rationalized by multiplying the numerator and denominator by :

giving the same final simplified result as we obtained earlier.

 

Example 3:

Simplify

solution:

No common factors can be found to cancel between the numerator and denominator here. Also, the square roots which appear here obviously cannot be simplified further. So, all that’s left to do to simplify this expression is to rationalize the denominator. This is accomplished by multiplying the numerator and denominator by , giving:

as the final result (since there are clearly no common factors that can be cancelled between the numerator and denominator of this final expression). Notice that we used brackets explicitly to ensure that the entire numerator was multiplied by in the first step here.