FreeAlgebra                             Tutorials! Home Polynomials Finding the Greatest Common Factor Factoring Trinomials Absolute Value Function A Summary of Factoring Polynomials Solving Equations with One Radical Term Adding Fractions Subtracting Fractions The FOIL Method Graphing Compound Inequalities Solving Absolute Value Inequalities Adding and Subtracting Polynomials Using Slope Solving Quadratic Equations Factoring Multiplication Properties of Exponents Completing the Square Solving Systems of Equations by using the Substitution Method Combining Like Radical Terms Elimination Using Multiplication Solving Equations Pythagoras' Theorem 1 Finding the Least Common Multiples Multiplying and Dividing in Scientific Notation Adding and Subtracting Fractions Solving Quadratic Equations Adding and Subtracting Fractions Multiplication by 111 Adding Fractions Multiplying and Dividing Rational Numbers Multiplication by 50 Solving Linear Inequalities in One Variable Simplifying Cube Roots That Contain Integers Graphing Compound Inequalities Simple Trinomials as Products of Binomials Writing Linear Equations in Slope-Intercept Form Solving Linear Equations Lines and Equations The Intercepts of a Parabola Absolute Value Function Solving Equations Solving Compound Linear Inequalities Complex Numbers Factoring the Difference of Two Squares Multiplying and Dividing Rational Expressions Adding and Subtracting Radicals Multiplying and Dividing Signed Numbers Solving Systems of Equations Factoring Out the Opposite of the GCF Multiplying Special Polynomials Properties of Exponents Scientific Notation Multiplying Rational Expressions Adding and Subtracting Rational Expressions With Unlike Denominators Multiplication by 25 Decimals to Fractions Solving Quadratic Equations by Completing the Square Quotient Rule for Exponents Simplifying Square Roots Multiplying and Dividing Rational Expressions Independent, Inconsistent, and Dependent Systems of Equations Slopes Graphing Lines in the Coordinate Plane Graphing Functions Powers of Ten Zero Power Property of Exponents The Vertex of a Parabola Rationalizing the Denominator Test for Factorability for Quadratic Trinomials Trinomial Squares Solving Two-Step Equations Solving Linear Equations Containing Fractions Multiplying by 125 Exponent Properties Multiplying Fractions Adding and Subtracting Rational Expressions With the Same Denominator Quadratic Expressions - Completing Squares Adding and Subtracting Mixed Numbers with Different Denominators Solving a Formula for a Given Variable Factoring Trinomials Multiplying and Dividing Fractions Multiplying and Dividing Complex Numbers in Polar Form Power Equations and their Graphs Solving Linear Systems of Equations by Substitution Solving Polynomial Equations by Factoring Laws of Exponents index casa mÃ­o Systems of Linear Equations Properties of Rational Exponents Power of a Product and Power of a Quotient Factoring Differences of Perfect Squares Dividing Fractions Factoring a Polynomial by Finding the GCF Graphing Linear Equations Steps in Factoring Multiplication Property of Exponents Solving Systems of Linear Equations in Three Variables Solving Exponential Equations Finding the GCF of a Set of Monomials

Try the Free Math Solver or Scroll down to Tutorials!

 Depdendent Variable

 Number of equations to solve: 23456789
 Equ. #1:
 Equ. #2:

 Equ. #3:

 Equ. #4:

 Equ. #5:

 Equ. #6:

 Equ. #7:

 Equ. #8:

 Equ. #9:

 Solve for:

 Dependent Variable

 Number of inequalities to solve: 23456789
 Ineq. #1:
 Ineq. #2:

 Ineq. #3:

 Ineq. #4:

 Ineq. #5:

 Ineq. #6:

 Ineq. #7:

 Ineq. #8:

 Ineq. #9:

 Solve for:

 Please use this form if you would like to have this math solver on your website, free of charge. Name: Email: Your Website: Msg:

# Simplifying Square Roots

An expression containing a square root is considered to be as simple as possible when the expression inside the square root is as simple or small as possible. The reduction of the contents inside the square root is accomplished (when possible) by a very straightforward strategy:

(i) Factor the expression inside the square root completely. Write factors which are perfect squares as explicit squares.

(ii) Use the property that the square root of a product is equal to the product of the square roots of the factors to rewrite the square root from step (i) as a product of square roots of factors which are perfect squares and a single square root of an expression which contains no perfect square factors. The pattern is: where u, v, etc. are perfect squares, and w is an expression containing no perfect square factors. (This may seem a bit abstract, but the meaning of this pattern should become more obvious after you have studied a few of the examples below. It is important in mathematics not only to study specific examples of a type of operation, but to eventually understand an overall general strategy or pattern for similar types of problems.

(iii) Replace the square roots of perfect squares by factors which are not square roots using the property We now illustrate this general strategy with a series of specific examples.

Example 1:

Simplify solution:

There is a strong temptation here to simply take the square root term by term to get However, you should see immediately that the first step violates the previously stated properties of radicals and so is invalid. The only way we can simplify this expression is if we are able to first factor it into a product in which one or more factors are perfect squares. Here, the expression x 2 + y 2 cannot be factored using any of the techniques available to us, and so no progress can be made as far as simplifying this square root. We are forced to conclude that the given expression is already in simplest radical form, and nothing can be done to reduce the expression inside the radical to a simpler algebraic form.

Example 2:

Simplify solution:

There seems to be a lot of perfect squares here – in fact, x 4, 9, and x 2 are all perfect squares. However, only perfect square factors of the entire expression in the radical are of any use to us here. Proceeding using the methods for factoring algebraic expressions that were covered in detail earlier in these notes, we get x 4 + 9x 2 = x 2(x 2 + 9) as the most complete factorization possible. Thus Again, the expression left in the square root in this last line cannot be factored further at all, and so we cannot extract any further perfect square factors inside the square root to allow further simplification of the square root. Thus must be the final answer here.

 All Right Reserved. Copyright 2005-2022