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Scientific Notation
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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
Decimals
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
Equations
   

Simplifying Radicals Involving Variables

To simplify radicals involving variables, we must recognize exponential expressions that are perfect squares, perfect cubes, and so on. The expressions x2, w4, y8, z14, and x50 are perfect squares because they are squares of variables with integral powers. Any even power of a variable is a perfect square. If we assume the variables represent positive numbers, we can write

Helpful hint

If you use exponential notation, then it is clear why the square root takes half of the exponent:

 

Note that when we find the square root, the result has one-half of the original exponent.

 

Example 1

Radicals with variables

Simplify each expression. Assume all variables represent positive real numbers.

Solution

a) Use the product rule to place all perfect squares under the first radical symbol and the remaining factors under the second:

Factor out the perfect squares.
  Product rule for radicals
   
Product rule for radicals
   
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