Radicals and Absolute Value Flashcards

Vocabulary terms covering basic properties of absolute values, radical simplification, and solving radical/absolute value equations based on pre-calculus lecture notes.

Absolute Value

The distance of a number from 0 on a number line, denoted as |a|. It is always non-negative.

Magnitude

The size of a number indicating its distance from 0, regardless of its sign.

Principal Square Root

The non-negative square root of a non-negative real number a, written as \sqrt{a}. For any real number x, \sqrt{x^2} = |x|.

Mixed Radical

A radical expression that includes a coefficient and a radical part, such as 3\sqrt{7}, where 3 is the coefficient.

Entire Radical

A radical where the entire value is contained under the radical sign with no coefficient other than 1, such as \sqrt{63}.

Rationalizing the Denominator

The process of converting a fraction with a radical in the denominator into an equivalent fraction with an integer denominator.

Conjugate

A binomial expression formed by changing the sign between two terms. For a + \sqrt{b}, the conjugate is a - \sqrt{b}. Multiplying a binomial by its conjugate results in a rational number.

Radical Equation

An equation containing at least one radical where the radicand includes a variable.

Extraneous Root

A solution obtained through algebraic methods (like squaring both sides) that does not satisfy the original equation.

Critical Point

In absolute value functions y = |f(x)|, the x-intercept where the graph changes direction and the definition of the piecewise function transitions.

Piecewise Notation

A way to define a function using different rules for different intervals of its domain. Example: |x| = \begin{cases} x, & \text{if } x \geq 0 \ -x, & \text{if } x < 0 \end{cases}.

Absolute Value Function

A function of the form y = |f(x)|. Its graph results from reflecting any part of the graph of y = f(x) that lies below the x-axis into the region above the x-axis.

Index

The small number in the crook of the radical symbol that indicates the root to be taken (e.g., n in \sqrt[n]{x}).

Radicand

The expression or number located inside the radical symbol.

Like Radicals

Radicals that have the same radicand and the same index, allowing them to be added or subtracted.