Radicals in Algebra Flashcards

Questions and Answers

What is a rational expression?

An expression involving a radical sign.

An algebraic expression written as a ratio. (correct)

An equation containing one or more rational expressions.

A binomial formed by taking the opposite of the second term.

How can you turn an unlike radical into a like radical?

By multiplying the conjugate.

What is a conjugate?

A binomial formed by taking the opposite of the second term of a binomial.

What is a radical equation?

An equation that has a variable in a radicand or has a variable with a rational exponent. Signup and view all the answers

What is a rational equation?

An equation containing one or more rational expressions. Signup and view all the answers

What is an extraneous solution?

A solution that does not satisfy the original equation. Signup and view all the answers

What defines a square root function?

The independent variable is contained within a square root. Signup and view all the answers

What are radical expressions?

Expressions involving a radical sign. Signup and view all the answers

What are like radicals?

Radical expressions that have the same index and the same radicand. Signup and view all the answers

What does it mean to rationalize a denominator?

To remove irrational quantities from the denominator. Signup and view all the answers

What is the quadratic formula?

x = -b ± √(b² - 4ac) / 2a. Signup and view all the answers

What is the discriminant in the quadratic formula?

The expression under the radical, b squared minus four a times c. Signup and view all the answers

What are complex numbers?

Numbers in the form a + bi, where b ≠ 0 are imaginary and b = 0 are real. Signup and view all the answers

What are imaginary numbers?

Numbers in the form a + bi when b ≠ 0. Signup and view all the answers

What defines rational numbers?

Numbers that can be expressed as a fraction. Signup and view all the answers

What defines irrational numbers?

Numbers that cannot be expressed as a fraction. Signup and view all the answers

What is a radicand?

The number inside the radical. Signup and view all the answers

What is the index in a radical expression?

The number outside the radical, indicating the degree. Signup and view all the answers

What is the conjugate of a + bi?

a - bi. Signup and view all the answers

What does i represent?

√-1. Signup and view all the answers

What does i² equal?

-1. Signup and view all the answers

What does i³ equal?

-i. Signup and view all the answers

What does i⁴ equal?

<ol> <li></li> </ol> Signup and view all the answers

How is the quadratic formula commonly represented?

-b ± √(b² + 4ac) / 2a. Signup and view all the answers

What does it mean if the discriminant is negative?

There are 2 imaginary conjugates. Signup and view all the answers

What does a positive square indicate?

There are 2 real rational roots. Signup and view all the answers

What does a positive non-square indicate?

There are 2 real irrational roots. Signup and view all the answers

Study Notes

Key Terms in Algebra

Rational Expression: An algebraic expression represented as a ratio of two polynomial expressions.
Unlike Radicals: Can be transformed into like radicals by multiplying with their conjugate.
Conjugate: A binomial created by changing the sign of the second term in a binomial expression.
Radical Equation: An equation containing a variable within a radicand or featuring a variable with a rational exponent.
Rational Equation: An equation that involves one or more rational expressions.
Extraneous Solution: A solution that fails to satisfy the original equation after substitution.
Square Root Function: A function where the independent variable is situated within a square root.
Radical Expressions: Any expression that includes a radical sign (√).
Like Radicals: Radical expressions sharing the same index and radicand.
Rationalize: The process of eliminating irrational numbers from denominators in expressions.

Important Mathematical Concepts

Quadratic Formula: The solution for quadratic equations is given by ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ).
Discriminant: The part of the quadratic formula under the square root, calculated as ( b^2 - 4ac ).

Types of Numbers

Complex Numbers: Expressed in the form ( a + bi ), where they are imaginary if ( b \neq 0 ) and real if ( b = 0 ).
Imaginary Numbers: Formulated as ( a + bi ) with ( b \neq 0 ).
Rational Numbers: Numbers that can be expressed as the quotient of two integers (fractions).
Irrational Numbers: Numbers that cannot be represented as fractions.

Radicals and Their Properties

Radicand: The number situated inside a radical symbol.
Index: Indicates the degree of the root outside the radical (e.g., square root has an index of 2).

Properties of Imaginary Unit ( i )

Conjugate of ( a + bi ): Formed as ( a - bi ).
( i ): Defined as ( \sqrt{-1} ).
( i^2 ): Equals -1.
( i^3 ): Equals -i.
( i^4 ): Equals 1.

Roots and Their Nature

Discriminant is Negative: Leads to two imaginary conjugate roots.
Positive Square: Results in two real rational roots.
Positive but Not a Square: Yields two real irrational roots.

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Description

Explore fundamental concepts of algebra focusing on radicals with these flashcards. Each card provides definitions for key terms like rational expressions and radical equations to enhance your understanding. Perfect for students looking to master algebraic expressions and equations involving radicals.