Algebra Tutorials!    
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Scientific Notation
Notation and Symbols
Linear Equations and Inequalities in One Variable
Graphing Equations in Three Variables
What the Standard Form of a Quadratic can tell you about the graph
Simplifying Radical Expressions Containing One Term
Adding and Subtracting Fractions
Multiplying Radical Expressions
Adding and Subtracting Fractions
Multiplying and Dividing With Square Roots
Graphing Linear Inequalities
Absolute Value Function
Real Numbers and the Real Line
Monomial Factors
Raising an Exponential Expression to a Power
Rational Exponents
Multiplying Two Fractions Whose Numerators Are Both 1
Multiplying Rational Expressions
Building Up the Denominator
Adding and Subtracting Decimals
Solving Quadratic Equations
Scientific Notation
Like Radical Terms
Graphing Parabolas
Subtracting Reverses
Solving Linear Equations
Dividing Rational Expressions
Complex Numbers
Solving Linear Inequalities
Working with Fractions
Graphing Linear Equations
Simplifying Expressions That Contain Negative Exponents
Rationalizing the Denominator
Estimating Sums and Differences of Mixed Numbers
Algebraic Fractions
Simplifying Rational Expressions
Linear Equations
Dividing Complex Numbers
Simplifying Square Roots That Contain Variables
Simplifying Radicals Involving Variables
Compound Inequalities
Factoring Special Quadratic Polynomials
Simplifying Complex Fractions
Rules for Exponents
Finding Logarithms
Multiplying Polynomials
Using Coordinates to Find Slope
Variables and Expressions
Dividing Radicals
Using Proportions and Cross
Solving Equations with Radicals and Exponents
Natural Logs
The Addition Method
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Multiplying and Dividing With Square Roots

We have already looked in some detail at multiplication and division with numerical square roots. The rules for multiplication and division with algebraic square roots are exactly the same as those for numerical square roots – in fact, we stated those rules there in an algebraic form:

property 1


property 2

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

When expressions involving square roots are to be multiplied or divided, just use the above rules to combine square roots as necessary. Then simplify the resulting expression as much as possible using methods of simplification already described and illustrated extensively in the preceding documents in this series.

The examples following below illustrate this general strategy, but also provide some examples for your own practice.


Example 1:



Using property (1) above,

as the final answer.


Example 2:



Using property (1) above, we get


Example 3:



You could regard this problem as a product

and use the procedures illustrated in the previous examples. However, an even faster way to get the answer is to distribute the power 2 over all factors in the brackets:

as the final answer. Here we have used the basic fact that´.


Example 4:



as the final simplified answer.


Example 5:



This is a product of three square root factors, and so does not fit property (1) at the beginning of this document precisely. However, we can apply property (1) in a stepwise fashion


which now does match property (1). Continuing

as the final simplified result.

The obvious implication of this stepwise application of property (1) is that property (1) can be extended to products of any number of square root factors.

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