Algebra & Arithmetic Quiz Bee Preparation

Vocabulary and formulas covering Exponents, Radicals, Linear Equations, Systems of Equations, and Quadratic Equations for Quiz Bee preparation.

Product Rule (Exponents)

$$a^m \cdot a^n = a^{m+n}$$

Quotient Rule (Exponents)

$$a^m / a^n = a^{m-n}$$

Power of a Power

$$(a^m)^n = a^{m \cdot n}$$

Power of a Product

$$(ab)^n = a^n b^n$$

Power of a Quotient

$$(a/b)^n = a^n / b^n$$

Zero Exponent

$a^0 = 1$ (where $a \neq 0$)

Negative Exponent

$$a^{-n} = \frac{1}{a^n}$$

Rational Exponent

$$a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$$

Radicand

The expression $a$ inside the radical symbol $\sqrt[n]{a}$

Index

The number $n$ in the radical expression $\sqrt[n]{a}$, indicating the root being taken

Nested Radicals

$$\sqrt[m]{\sqrt[n]{a}} = \sqrt[m \cdot n]{a}$$

Conjugate

The expression $a - \sqrt{b}$ used to rationalize $a + \sqrt{b}$, because $(a + \sqrt{b})(a - \sqrt{b}) = a^2 - b$

Slope-Intercept Form

$y = mx + b$ (where $m$ is the slope and $b$ is the y-intercept)

Standard Form (Linear)

$Ax + By = C$ (where $A, B, C \in \mathbf{Z}$ and $A \geq 0$)

Point-Slope Form

$$y - y_1 = m(x - x_1)$$

Slope Formula

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

Parallel Lines (Slopes)

Lines that have equal slopes ($m_1 = m_2$)

Perpendicular Lines (Slopes)

Lines whose slopes are negative reciprocals ($m_1 \cdot m_2 = -1$)

Distance Formula

$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

Midpoint Formula

$$M = (\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})$$

Consistent & Independent System

A system where $\frac{A_1}{A_2} \neq \frac{B_1}{B_2}$, resulting in exactly one solution and intersecting lines

Consistent & Dependent System

A system where $\frac{A_1}{A_2} = \frac{B_1}{B_2} = \frac{C_1}{C_2}$, resulting in infinitely many solutions and coincident lines

Inconsistent System

A system where $\frac{A_1}{A_2} = \frac{B_1}{B_2} \neq \frac{C_1}{C_2}$, resulting in no solution and parallel lines

Cramer's Rule Determinant (D)

For a $2 \times 2$ system, $D = A_1 B_2 - A_2 B_1$

Standard Form (Quadratic)

$ax^2 + bx + c = 0$ (where $a \neq 0$)

Quadratic Formula

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Discriminant (D)

$$D = b^2 - 4ac$$

Vieta's Formulas (Sum of Roots)

$$x_1 + x_2 = -\frac{b}{a}$$

Vieta's Formulas (Product of Roots)

$$x_1 \cdot x_2 = \frac{c}{a}$$

Quadratic Vertex (h)

$$h = -\frac{b}{2a}$$

Quadratic Vertex (k/Extreme Value)

$$k = \frac{4ac - b^2}{4a}$$