Manipulating Expressions that Contain Radicals, and Rational Exponents.

You hear radicals, and you get a headache. You see an exponent that is a fraction and you panic! Well, there is no reason for that frustration. After you have gone through this lesson; you will discover that you have all the skills to tackle any expression with radicals, or fractions in the exponents. You will explore patterns that are very telling for the reason behind all these little rules. In fact you will enjoy the challenge to simplify them!

Lesson's Content

Lesson In PDF Format (no animations)


Lesson's Glossary

Algebraic expression: A given set of letters called variables, and real numbers called constants that are combined using addition, subtraction, multiplication, division and/or exponentiation.

Constant: It is a number or value that remains always the same. Never changes.

Exponent: It is a raised number representing the repeated multiplication of a given factor.

Perfect square trinomial: A trinomial generated by the product of two equal binomials.

Expression: Any combination of numbers and operations without the = sign. 

Factors: All whole numbers that are multiplied together to yield another number.

Like terms: Terms that have the same combination of variables to the same power as factors.

Negative square root: The opposite root to the principal root of the square root.

Operation: Any action we perform on one or two numbers to produce a new number. Most common ones are addition, subtraction, multiplication, division, square roots, powers, and so on.

Polynomial: An algebraic statement with one or more terms. Word comes from “poly” which means many.

Power: Exponent of a number or variable.

Principal root: Given a number with two roots, the positive root is the principal and its given by the radical sign.

Rational exponent: Given that the nth root of x is a real number and m is an integer, we have that x1/n = n√x and xm/n = nxm= (na)n.

Rational number: Any number that may written as a fraction; including whole numbers (written as fractions with 1 as denominator) and decimals that truncate or repeat (may be expressed as fractions).

Rationalize the denominator: The process of changing the denominator of a fraction from radical expression to rational number without changing the value of the expression.

Real number: Any number rational or irrational.
With exception of the imaginary numbers all numbers are real numbers.

Reciprocal: Multiplicative inverse of a number different than zero. The multiplication of a number and its inverse is 1.

Subtraction: Adding the opposite

Term: A form of grouping one or more numerical and/or variable factors by means of multiplication and division. Addition and subtraction symbols separate terms.

Variable: A letter used to represent a number. When the variable is part of an equation, it is possible to find the value for which it stands for by solving the equation. This is the solution (s) of the equation.

Variable expression: Mathematical phrase with at least one variable in it.

Radical: The symbol that encloses a square root or any other given root.

Radicand: The number that is under the radicand sign.


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