Algebra Tutorials!  
Monday 15th of July
Rotating a Parabola
Multiplying Fractions
Finding Factors
Miscellaneous Equations
Mixed Numbers and Improper Fractions
Systems of Equations in Two Variables
Literal Numbers
Adding and Subtracting Polynomials
Subtracting Integers
Simplifying Complex Fractions
Decimals and Fractions
Multiplying Integers
Logarithmic Functions
Multiplying Monomials
The Square of a Binomial
Factoring Trinomials
The Pythagorean Theorem
Solving Radical Equations in One Variable
Multiplying Binomials Using the FOIL Method
Imaginary Numbers
Solving Quadratic Equations Using the Quadratic Formula
Solving Quadratic Equations
Order of Operations
Dividing Complex Numbers
The Appearance of a Polynomial Equation
Standard Form of a Line
Positive Integral Divisors
Dividing Fractions
Solving Linear Systems of Equations by Elimination
Multiplying and Dividing Square Roots
Functions and Graphs
Dividing Polynomials
Solving Rational Equations
Use of Parentheses or Brackets (The Distributive Law)
Multiplying and Dividing by Monomials
Solving Quadratic Equations by Graphing
Multiplying Decimals
Use of Parentheses or Brackets (The Distributive Law)
Simplifying Complex Fractions 1
Adding Fractions
Simplifying Complex Fractions
Solutions to Linear Equations in Two Variables
Quadratic Expressions Completing Squares
Dividing Radical Expressions
Rise and Run
Graphing Exponential Functions
Multiplying by a Monomial
The Cartesian Coordinate System
Writing the Terms of a Polynomial in Descending Order
Quadratic Expressions
Solving Inequalities
Solving Rational Inequalities with a Sign Graph
Solving Linear Equations
Solving an Equation with Two Radical Terms
Simplifying Rational Expressions
Intercepts of a Line
Completing the Square
Order of Operations
Factoring Trinomials
Solving Linear Equations
Solving Multi-Step Inequalities
Solving Quadratic Equations Graphically and Algebraically
Collecting Like Terms
Solving Equations with Radicals and Exponents
Percent of Change
Powers of ten (Scientific Notation)
Comparing Integers on a Number Line
Solving Systems of Equations Using Substitution
Factoring Out the Greatest Common Factor
Families of Functions
Monomial Factors
Multiplying and Dividing Complex Numbers
Properties of Exponents
Multiplying Square Roots
Adding or Subtracting Rational Expressions with Different Denominators
Expressions with Variables as Exponents
The Quadratic Formula
Writing a Quadratic with Given Solutions
Simplifying Square Roots
Adding and Subtracting Square Roots
Adding and Subtracting Rational Expressions
Combining Like Radical Terms
Solving Systems of Equations Using Substitution
Dividing Polynomials
Graphing Functions
Product of a Sum and a Difference
Solving First Degree Inequalities
Solving Equations with Radicals and Exponents
Roots and Powers
Multiplying Numbers
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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:



The expression inside the square root is nearly completely factored already, but we do need to write it so that perfect square factors show up more explicitly to complete step (i) in applying the general strategy given above. We have

50 = 2 · 5 · 5 = 2 · 5 2 and x 13 = x 12 · x = (x 6) 2 · x

Thus our original square root can be rewritten as

after rearranging the order of the factors. Now we can do step (ii) of the strategy: separate the perfect square factors in this square root into individual square root factors.
now we can apply step (iii) of the strategy to the first two factors …
to get our final result.

Notice that the remaining square root factor in this final expression, , contains no further perfect square factors, and so there is no way to reduce the expression inside this remaining square root any further. Thus, our final answer is:

Even though you can probably do the first step of this strategy – the factorization to show perfect square factors – without writing down a lot of intermediate steps, we suggest that you do write down enough of the details to reliably keep track of what you are doing. It is particularly important to ensure that the perfect squares that you identify are accurate, and the best way to do that is to show at least as much work as we did in the solution to Example 1 above. For powers of symbols, even powers are always perfect squares. Odd powers are a product of a perfect square and the first power of that symbol.


Example 2:



First, we factor the expression inside the square root to display all possible perfect square factors explicitly. Methods for ensuring a complete factorization of numbers into a product of prime factors have been covered elsewhere in these notes.

400 = 2 · 200 = 2 · 2 · 100 = 2 · 2 · 2 · 50 = 2 · 2 · 2 · 2 · 25 = 2 · 2 · 2 · 2 · 5 · 5 = 2 4 · 5 2 = (2 2 ) 2 · 5 2


y 7 = y 6 · y = (y 3 ) 2 y


as the final answer.


Example 3:




32 = 2 5 = (2 2 ) 2 · 2


ab 3c 4 = a · b 2 · b · (c 2) 2 = b 2(c 2) 2 · ab


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