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Pythagoras' Theorem 1
Finding the Least Common Multiples
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Multiplication by 111
Adding Fractions
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Multiplication by 50
Solving Linear Inequalities in One Variable
Simplifying Cube Roots That Contain Integers
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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
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Complex Numbers
Factoring the Difference of Two Squares
Multiplying and Dividing Rational Expressions
Adding and Subtracting Radicals
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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
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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
 
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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.

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