RadF - Radical FunctionsModule Overview

Radical Functions Module Overview

Introduction

Radical functions are functions that involve radicals, which can be written as rational exponents.

Radical Expressions are expressions involving radicals, and when written with radicals this is known as radical form. Radical expressions can also be written with rational exponents (known as exponential form). These radicals contain an index (sometimes also referred to as a "root"). Roots and powers are opposite operations. These similarities are also in the operations of addition and subtraction and the operations of multiplication and division.

For an integer, n, greater than 1, if  $b^n=a$ , then b is an nth root of a. An nth root of a, written as  $\sqrt[n]{a}$ , where n is the index of the radical. If n is odd, then a has one real nth root. If n is even, then a has two real nth roots a > 0, one real nth root if a = 0, and no real nth roots if a < 0. When we are graphing radical functions, we will specifically explore the square roots (  $y=\sqrt[]{x}$  ) and cube roots ( $y=\sqrt[3]{x}$  ). In this unit, we'll explore some more specifics of radical functions, operations on radical functions (written in radical form and in rational exponent form), solving radical functions and inequalities, and then graphing radical functions and inequalities.

Essential Questions

• What are various properties of nth roots?
• What are various properties of rational exponents?
• How is a radical (rational exponent) equation solved?
• Why are all solutions not necessarily the solution to an equation?
• How can extraneous solutions be identified?
• How are square root and cube root functions graphed?

Radical Functions Key Terms

Algebra - The branch of mathematics that deals with relationships between numbers, utilizing letters and other symbols to represent specific sets of numbers, or to describe a pattern of relationships between numbers.

Coefficient - A number multiplied by a variable.

Equation - A number sentence that contains an equality symbol.

Expression - A mathematical phrase involving at least one variable and sometimes numbers and operation symbols.

Extraneous Solution(s) - A solution of the simplified form of the equation that does not satisfy the original equation.

Index of a Radical - The " n " in the nth root of a, written as $\sqrt[n]{a}$.

Inequality - Any mathematical sentence that contains the symbols > (greater than), < (less than), ≤ (less than or equal to), or ≥ (greater than or equal to).

Polynomial - A mathematical expression involving the sum of terms made up of variables to nonnegative integer powers and real-valued coefficients.

Radical Function - A function containing a root. The most common radical functions are the square root and cube root functions,

$f\left(x\right)=a\sqrt[]{x-h}+k\:and\:f\left(x\right)=a\sqrt[3]{x-h}+k$

Reciprocal - Two numbers whose product is one. For example, $\frac{m}{n}\cdot\frac{n}{m}=1$.

Variable - A letter or symbol used to represent a number.

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