Product Rule for Exponents
Decimal Numbers and Fractions
The Slope of a Line
Adding and Subtracting Square Roots
Factoring the Difference of Two Squares
Linear Systems of Equations with Infinitely Many Solutions
Axis of Symmetry and Vertices
Types of Linear Equations
Sum of Squares
Subtract Fractions with Unlike Denominators
Solving equations
Solving Exponential Equations
Multiplying by 715
Adding and Subtracting Functions
Negative and Fractional Powers
Graphing Linear Equations in Two Variables
Solving Equations That Contain Rational Expressions
Dividing Polynomials
Polynomials in Several Variables
Multiplying Polynomials
Adding and Subtracting Fractions
Solving Absolute Value Inequalities
Simplifying Complex Fractions
Evaluating Rational Functions
Product Rule for Radicals
Domain and Range of a Function
Solving Linear Equations
Dividing Whole Numbers by Fractions
Reducing Rational Expressions to Lowest Terms
Dividing Polynomials
Factoring by Substitution
Dividing a Polynomial by a Monomial
Linear Inequalities
Adding and Subtracting Complex Numbers
What the Vertex Form of a Quadratic can tell you about the graph
Finding x
Adding and Subtracting Fractions with Like Denominators
Adding and Subtracting Fractions
Solving Equations
Graphing Linear Equations
Greatest Common Factors
Exponential Functions
Methods for Solving Quadratic Equations
Factoring Trinomials with Leading Coefficient Not 1
Properties of Natural Logs
Steps for Solving Linear Equations
Multiplying Binomials
Factoring Trinomials
Adding and Subtracting Mixed Numbers with Different Denominators
Simplifying Complex Fractions
Sum or Difference of two Cubes
Multiplying by 858
Graphing Quadratic Equations
Rational Expressions
Graphing Vertical Lines
Dividing Fractions
Multiplying Numers
Multiplying Two Numbers Close to but greater than 100
Factoring Trinomials
Equivalent Fractions
Finding the Least Common Multiple
Factoring Rules
Laws of Exponents
Multiplying Polynomials
Dividing Rational Expressions
Evaluating Polynomial Functions
Equations Involving Rational Exponents
Adding and Subtracting Fractions
Factoring Polynomials by Finding the Greatest Common Factor
Rules for Integral Exponents
Rationalizing the Denominator
Ratios and Rates
Factoring Trinomials
Multiplying Polynomials
Point-Slope Form of a Line
Multiplying Decimals
Solving Right Triangles
Solving Equations with One Radical Term
Adding and Subtracting Mixed Numbers
Adding and Subtracting Polynomials
Division Property of Square and Cube Roots
Inverse Functions
Factoring Trinomials
Writing Percents as Fractions
Solving Equations with One Radical Term
Graphing Systems of Inequalities
Multiplying and Dividing Monomials
Roots - Radicals 2
Solving Linear Systems of Equations
Multiplying and Factoring
Solving Equations with Rational Expressions

Product Rule for Radicals

Recall that the power of a product rule is valid for rational exponents as well as integers. For example, the power of a product rule allows us to write

(4y)1/2 = 41/2 · y1/2 and (8 · 7)1/3 = 81/3 · 71/3.

These equations can be written using radical notation as

The power of a product rule (for the power 1/n) can be stated using radical notation. In this form the rule is called the product rule for radicals.


Product Rule for Radicals

The nth root of a product is equal to the product of the nth roots. In symbols,

 provided that all of the expressions represent real numbers.

The numbers 1, 4, 9, 16, 25, 49, 64, and so on are called perfect squares because they are the squares of the positive integers. If the radicand of a square root has a perfect square (other than 1) as a factor, the product rule can be used to simplify the radical expression. For example, the radicand of has 25 as a factor, so we can use the product rule to factor into a product of two square roots:

When simplifying a cube root, we check the radicand for factors that are perfect cubes: 8, 27, 64, 125, and so on. In general, when simplifying an nth root, we look for a perfect nth power as a factor of the radicand.


Example 1

Using the product rule to simplify radicals

Simplify each expression. Assume all variables represent positive numbers.


a) The radicand 4y has the perfect square 4 as a factor. So

b) The radicand 18 has a factor of 9. So

c) The radicand 56 in this cube root has the perfect cube 8 as a factor. So

d) The radicand in this fourth root has the perfect fourth power 16 as a factor. So