Exponent Rules, Factoring, and Distance Formula (Video Notes)

A set of vocabulary flashcards covering exponent rules, radical simplification, factoring methods, FOIL/grid methods, the distance formula, and related concepts from the video notes.

Product Rule for Exponents

When multiplying like bases, add the exponents: x^a * x^b = x^(a+b).

Negative Exponent

An exponent with a negative power; a^-n = 1 / a^n, the reciprocal of the base raised to the positive exponent.

Zero Exponent Rule

Any nonzero base raised to the 0 power equals 1: x^0 = 1 (x ≠ 0).

Quotient Rule for Exponents

When dividing like bases, subtract the exponents: x^a / x^b = x^(a-b).

Fractional Exponents

An exponent m/n means the nth root of the base raised to m: x^(m/n) = (x^m)^(1/n) = nth root of x^m.

Square Root Interpretation

x^(1/2) equals the square root of x.

Cube Root Interpretation

x^(1/3) equals the cube root of x.

Fractional Exponent Example (y^(2/5))

y^(2/5) means the fifth root of y^2 (or (y^2)^(1/5)).

Radical Simplification

Extract perfect powers from under a radical to the outside; remaining under-root factors stay inside the radical.

FOIL Method

First, Outer, Inner, Last; a method to multiply binomials.

Grid Method

An alternative to FOIL using a grid to organize multiplication of binomials.

Difference of Squares

A^2 − B^2 factors as (A − B)(A + B).

Factoring Quadratics with a=1

For ax^2 + bx + c with a=1, factor as (x + m)(x + n) where m and n multiply to c and add to b.

AC Method (Factoring by Grouping)

Multiply a and c and find two numbers that multiply to ac and sum to b to rewrite the middle term.

Factoring by Pairs

Factor quadratics by grouping terms into pairs to factor into a product of binomials.

Zeros from Factoring / Graphing

Set each factor equal to zero to find the roots; graphing can show the zeros of a quadratic.

Distance Formula

Distance between (x1,y1) and (x2,y2) is sqrt((x2−x1)^2 + (y2−y1)^2).

Pythagorean Theorem Connection

In a right triangle, a^2 + b^2 = c^2; foundational for the distance formula via base and height.

FOIL vs Distribute Distinction

FOIL expands binomials; distribute applies to multiplying a sum by another expression; use FOIL for binomial products.