Exponential Functions
Linera Equations
Simple Trinomials as Products of Binomials
Laws of Exponents and Dividing Monomials
Solving Equations
Multiplying Polynomials
Multiplying and Dividing Rational Expressions
Solving Systems of Linear Inequalities
Mixed-Number Notation
Linear Equations and Inequalities in One Variable
The Quadratic Formula
Fractions and Decimals
Graphing Logarithmic Functions
Multiplication by 111
Solving Systems of Equations - Two Lines
Solving Nonlinear Equations by Factoring
Solving Linear Systems of Equations by Elimination
Rationalizing the Denominator
Simplifying Complex Fractions
Factoring Trinomials
Linear Relations and Functions
Axis of Symmetry and Vertices
Equations Quadratic in Form
The Appearance of a Polynomial Equation
Subtracting Reverses
Non-Linear Equations
Exponents and Order of Operations
Factoring Trinomials by Grouping
Factoring Trinomials of the Type ax 2 + bx + c
The Distance Formula
Invariants Under Rotation
Multiplying and Dividing Monomials
Solving a System of Three Linear Equations by Elimination
Multiplication by 25
Powers of i
Solving Quadratic and Polynomial Equations
Slope-intercept Form for the Equation of a Line
Equations of Lines
Square Roots
Integral Exponents
Product Rule for Radicals
Solving Compound Linear Inequalities
Axis of Symmetry and Vertices
Multiplying Rational Expressions
Reducing Rational Expressions
Properties of Negative Exponents
Numbers, Factors, and Reducing Fractions to Lowest Terms
Solving Quadratic Equations
Factoring Completely General Quadratic Trinomials
Solving a Formula for a Given Variable
Factoring Polynomials
Decimal Numbers and Fractions
Multiplication Properties of Exponents
Multiplying Fractions
Multiplication by 50

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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