Algebraic Manipulation Review

Flashcards covering key vocabulary from the Algebraic Manipulation lecture notes, including equivalent expressions, algebraic properties, exponents, radicals, and fraction operations.

Equivalent Expressions

Expressions that give equal outputs for specific inputs (demonstrated by showing) or for all inputs (demonstrated by justifying).

Show (Equivalency)

A way to demonstrate that specific inputs cause equivalency by choosing input values and evaluating both expressions.

Justify (Equivalency)

An algebraic proof that demonstrates equivalency for ALL inputs using algebraic properties.

Distributing

An algebraic process where a factor outside parentheses is multiplied by each term inside the parentheses.

Combining Like-Terms

An algebraic process of grouping and simplifying terms that have the same variable(s) raised to the same power(s).

Factoring (Standard Quadratics)

The process of breaking down a quadratic expression (e.g., ax2 + bx + c) into a product of linear factors.

Factoring (Difference of Perfect Squares)

Factoring an expression of the form x2 - y2 into (x + y)(x - y).

Factoring (Greatest Common Factor - GCF)

Taking out the largest common factor from all terms in an expression.

Exponent

The index or power in an exponential expression, indicating how many times the base is multiplied by itself.

Base (Exponents)

The number or variable that is being multiplied by itself in an exponential expression.

Property of 0 (Exponents)

A base raised to the power of 0 equals 1 (b0 = 1).

Property of Negatives (Exponents)

A base raised to a negative exponent is equal to 1 divided by the base raised to the positive exponent (b-n = 1/bn).

Product Property (Exponents)

When multiplying bases with the same base, add the exponents (am * an = a(m+n)).

Power of a Power (Exponents)

When raising a power to another power, multiply the exponents ((am)n = a(mn)).

Quotient Property (Exponents)

When dividing bases with the same base, subtract the exponents (am / an = a(m-n)).

Radical Expression

An expression that uses a radical symbol (√) to indicate the nth root of a number.

Nth Root

The value that, when raised to the nth power, gives the number under the radical sign.

Rationalize the Denominator

A process to convert a denominator containing a radical into a whole number, typically by multiplying the fraction by a form of 1.

Fraction Exponents

A way to rewrite radicals, where the power is the numerator and the root is the denominator of the exponent (x(a/b) = b√(xa)).

Cancelling Common Factors

Simplifying algebraic fractions by dividing out identical factors present in both the numerator and the denominator.