Elimination Using Multiplication
Prime Factors
Equations Involving Rational Exponents
Working with Percentages and Proportions
Rational Expressions
Interval Notation and Graphs
Simplifying Complex Fractions
Dividing Whole Numbers with Long Division
Solving Compound Linear Inequalities
Raising a Quotient to a Power
Solving Rational Equations
Solving Inequalities
Adding with Negative Numbers
Quadratic Inequalities
Dividing Monomials
Using the Discriminant in Factoring
Solving Equations by Factoring
Subtracting Polynomials
Cube Root
The Quadratic Formula
Multiply by the Reciprocal
Relating Equations and Graphs for Quadratic Functions
Multiplying a Polynomial by a Monomial
Calculating Percentages
Solving Systems of Equations using Substitution
Comparing Fractions
Solving Equations Containing Rational Expressions
Factoring Polynomials
Negative Rational Exponents
Roots and Radicals
Intercepts Given Ordered Pairs and Lines
Factoring Polynomials
Solving Linear Inequalities
Mixed Expressions and Complex Fractions
Solving Equations by Multiplying or Dividing
The Addition Method
Finding the Equation of an Inverse Function
Solving Compound Linear Inequalities
Multiplying and Dividing With Square Roots
Exponents and Their Properties
Equations as Functions
Factoring Trinomials
Solving Quadratic Equations by Completing the Square
Dividing by Decimals
Lines and Equations
Simplifying Complex Fractions
Graphing Solution Sets for Inequalities
Standard Form for the Equation of a Line
Checking Division with Multiplication
Elimination Using Addition and Subtraction
Complex Fractions
Multiplication Property of Equality
Solving Proportions Using Cross Multiplication
Product and Quotient of Functions
Quadratic Functions
Solving Compound Inequalities
Operating with Complex Numbers
Equivalent Fractions
Changing Improper Fractions to Mixed Numbers
Multiplying by a Monomial
Solving Linear Equations and Inequalities Graphically
Dividing Polynomials by Monomials
Multiplying Cube Roots
Operations with Monomials
Properties of Exponents
Mixed Numbers and Improper Fractions
Equations Quadratic in Form
Simplifying Square Roots That Contain Whole Numbers
Dividing a Polynomial by a Monomial
Writing Numbers in Scientific Notation
Solutions to Linear Equations in Two Variables
Solving Linear Inequalities
Multiplying Two Mixed Numbers with the Same Fraction
Special Fractions
Solving a Quadratic Inequality
Parent and Family Graphs
Solving Equations with a Fractional Exponent
Evaluating Trigonometric Functions
Solving Equations Involving Rational Expressions
Laws of Exponents
Multiplying Polynomials
Vertical Line Test
Solving Inequalities with Fractions and Parentheses
Multiplying Polynomials
Solving Quadratic and Polynomial Equations
Extraneous Solutions
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Multiplying and Dividing With Square Roots

The following rules for multiplication and division with algebraic square roots are the same as those for numerical square roots:

  • Property 1:
  • Property 2:


where ‘a’ and ‘b’ stand for any valid mathematical expression.


Example 1:



We can simply go ahead and apply property (1), getting

But now, we need to recall that simplification of square roots involves finding perfect square factors in the square root, so there is no point in expanding the product of the two binomials in this square root. Instead, we check for the possibility of further factoring

6x 2 + 24x = 6x(x + 4)


18x 4 + 72x 3 = 18x 3(x + 4)



Example 2:

Expand and simplify


At the start, this example involves the square of a binomial. You can use either of the two approaches presented earlier for multiplying one binomial by another binomial. The result here is

as the final answer.

Two errors are commonly made with this type of problem:

(1) People sometimes just square each term of the binomial:

But, you know that this is not the correct way to expand what amounts to the product of two binomials.

(2) People sometimes forget the obvious simplification that , leaving their answer as

instead of


Example 3:

Expand and simplify


This example is very similar to the problem in Example 2 above – the product of two binomials, each of which contains a square root. We get

as the final result.


Example 4:

Expand and simplify



as the final result. The two middle terms in the second last line are identical but of opposite sign, and so cancel each other. This sort of pattern of binomial factors, where each binomial contains square roots but their product contains no square roots, is the basis for methods to rationalize denominators of fractions which have binomial denominators.


Example 5:



We can use either of two approaches here. We could start by using property (2) from the beginning of this document to write

and then simplify the fraction in the square root to get

as a final, simplified answer. This approach is quite short and efficient.

A second approach is to simplify the square roots first

and then rationalize the denominator:


and so

which is exactly the same final answer as we obtained with the first method.

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