Elimination Using Multiplication
Prime Factors
Equations Involving Rational Exponents
Working with Percentages and Proportions
Rational Expressions
Interval Notation and Graphs
Simplifying Complex Fractions
Dividing Whole Numbers with Long Division
Solving Compound Linear Inequalities
Raising a Quotient to a Power
Solving Rational Equations
Solving Inequalities
Adding with Negative Numbers
Quadratic Inequalities
Dividing Monomials
Using the Discriminant in Factoring
Solving Equations by Factoring
Subtracting Polynomials
Cube Root
The Quadratic Formula
Multiply by the Reciprocal
Relating Equations and Graphs for Quadratic Functions
Multiplying a Polynomial by a Monomial
Calculating Percentages
Solving Systems of Equations using Substitution
Comparing Fractions
Solving Equations Containing Rational Expressions
Factoring Polynomials
Negative Rational Exponents
Roots and Radicals
Intercepts Given Ordered Pairs and Lines
Factoring Polynomials
Solving Linear Inequalities
Mixed Expressions and Complex Fractions
Solving Equations by Multiplying or Dividing
The Addition Method
Finding the Equation of an Inverse Function
Solving Compound Linear Inequalities
Multiplying and Dividing With Square Roots
Exponents and Their Properties
Equations as Functions
Factoring Trinomials
Solving Quadratic Equations by Completing the Square
Dividing by Decimals
Lines and Equations
Simplifying Complex Fractions
Graphing Solution Sets for Inequalities
Standard Form for the Equation of a Line
Checking Division with Multiplication
Elimination Using Addition and Subtraction
Complex Fractions
Multiplication Property of Equality
Solving Proportions Using Cross Multiplication
Product and Quotient of Functions
Quadratic Functions
Solving Compound Inequalities
Operating with Complex Numbers
Equivalent Fractions
Changing Improper Fractions to Mixed Numbers
Multiplying by a Monomial
Solving Linear Equations and Inequalities Graphically
Dividing Polynomials by Monomials
Multiplying Cube Roots
Operations with Monomials
Properties of Exponents
Mixed Numbers and Improper Fractions
Equations Quadratic in Form
Simplifying Square Roots That Contain Whole Numbers
Dividing a Polynomial by a Monomial
Writing Numbers in Scientific Notation
Solutions to Linear Equations in Two Variables
Solving Linear Inequalities
Multiplying Two Mixed Numbers with the Same Fraction
Special Fractions
Solving a Quadratic Inequality
Parent and Family Graphs
Solving Equations with a Fractional Exponent
Evaluating Trigonometric Functions
Solving Equations Involving Rational Expressions
Laws of Exponents
Multiplying Polynomials
Vertical Line Test
Solving Inequalities with Fractions and Parentheses
Multiplying Polynomials
Solving Quadratic and Polynomial Equations
Extraneous Solutions
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Cube Root

Definition of Cube Root

To indicate a cube root, we use the same radical symbol we used for square roots, but we write a small 3 just above the on the radical symbol to indicate a cube root. The small 3 is called the index of the radical.

For example, the cube root of 64 is written


Definition — Cube Root

The cube root of a real number, a, is written If b is a real number and b3 = a, then

Example: because 43 = 64.

Here are some examples:

because 53 = 5 · 5 · 5 = 125

because 13 = 1 · 1 · 1 = 1.

= 1 because (-1)3 = (-1)(-1)(-1) = -1.

because (-2)3 = (-2)(-2)(-2) = -8.


To find the cube root of a number, we reverse the operation of cubing.

For example, to find we ask “What number cubed is 8?”.

The answer is 2.


The last two examples show that a cube root, unlike a square root, can have a negative radicand.

We can use geometry to provide a visual interpretation of a cube root.

For example, suppose a cube has a volume of 125 cubic inches. The length of each side is the cube root of the volume.

That is, the length of a side of the cube == 5 in.

A perfect cube is a number that has a rational cube root.

For example, 125 is a perfect cube because 53 = 125.

As we work with cube roots, we will find it useful to recognize perfect cubes and their cube roots.

Now we will summarize the relationship between cubes and cube roots.


Property — Cubes and Cube Roots

English Cubing and taking a cube root “undo” each other.

Algebra If a is a real number, then



Unlike square roots, there are no “principal” cube roots.

The sign of a cube root depends on the sign of its radicand.

Perfect Squares Principal Square Roots
03 = 0
13 = 1
23 = 4
33 = 9
43 = 16
53 = 25
63 = 36
73 = 49
83 = 64
93 = 81
103 = 100

Property — Cubes and Cube Roots

English Cubing and taking a cube root “undo” each other.

Algebra If a is a real number, then


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