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Operations with Radical Expressions

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More lessons on Basic Algebra

More examples of rationalizing the denominator

Math Worksheets

A series of free, online Basic Algebra Lessons.

In this lesson, we will learn

  • how to simplify radical expressions
  • how to add and subtract radical expressions
  • how to multiply and simplify radical expressions
  • how to multiply radical expressions that are binomials
  • how to divide radical expressions
  • how to rationalize the denominator of a rational expression

Simplifying Radical Expressions

Radical expressions are square roots of monomials, binomials, or polynomials. One way of simplifying radical expressions is to break down the expression into perfect squares multiplying each other. A perfect square, such as 4, 9, 16 or 25, has a whole number square root. Simplifying radical expressions becomes especially important in Geometry when solving formulas and in using the Pythagorean Theorem.

Simplifying Radical Expressions

Simplifying Radical Expressions Involving Variables - Example 1

Simplifying Radical Expressions Involving Variables - Example 2

Simplifying Radical Expressions Involving Variables - Example 3


Adding and Subtracting Radical Expressions

Adding and subtracting radical expressions is similar to combining like terms: if two terms are multiplying the same radical expression, their coefficients can be summed. Some problems will not start with the same roots but the terms can be added after simplifying one or both radical expressions. Adding like radicals appears later in Algebra and frequently in Geometry.

Adding and Subtracting Radical Expressions

This video explains how to add and subtract radicals.

Multiplying and Distributing Radical Expressions

When multiplying radical expressions, it is helpful to remember that only terms "inside" the radicals can be multiplied and only the terms "outside" the radical can be multiplied together. After multiplying radicals, check to see if any perfect squares can be factored out to simplify the radical expression. Multiplying radicals is used to rationalize radicals and in Geometry.

Multiplying Radical Expressions

Multiply and Simply a Radical Expression.

Multiply radical expressions that are in the form of binomials by using FOIL
Multiplying Binomials with Radicals

Dividing Radicals and Rationalizing the Denominator

To "rationalize" a fraction means to perform some operation to remove any radicals from the denominator. When dividing radicals (fractions are division), start by reducing and looking for perfect squares. If the denominator has a sum or difference of two different radicands, multiply the numerator and denominator by a conjugate to remove the radicals.

Divide and Simply a Radical Expression

How to Rationalize a Denominator of a radical expression
Rationalize the Denominator

This video shows how to rationalize the denominator of a fraction. It also shows a few basic examples as well as multiplying by the conjugate<


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You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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