Algebra Essentials Cheat Sheet

Master fundamental algebraic concepts including exponents, radicals, factoring, trigonometry, functions, and linear systems through clear formulas and step-by-step problem-solving.

Core Principles

• Exponent rules simplify powers with the same base.
• Radicals can be converted to fractional exponents and vice versa.
• Factoring involves finding expressions that multiply to a given polynomial.
• Trigonometry relates angles and sides in right triangles (SOH CAH TOA).
• A function maps each input to exactly one output.
• Linear functions have a constant rate of change (slope).
• Systems of equations involve finding values that satisfy multiple equations simultaneously.

Action Steps

• **Exponents/Radicals:** Apply rules for product, quotient, power, negative, and rational exponents. Convert between radical and fractional exponent forms.
• **Factoring:** Identify GCF, factor simple/complex trinomials, and use grouping.
• **Trigonometry:** Use SOH CAH TOA for side/angle calculations. Employ inverse trig functions for angles.
• **Functions:** Verify each input has one output. Evaluate functions by substitution.
• **Linear Functions:** Calculate slope, find perpendicular slopes, and determine intercepts.
• **Systems of Equations:** Solve using substitution or elimination methods.
• **Word Problems:** Translate scenarios into algebraic expressions, equations, or systems.

Formulas

• $a^m \cdot a^n = a^{m+n}$
• $a^m / a^n = a^{m-n}$
• \((a^m)^n = a^{m \cdot n}\)
• $a^0 = 1$
• $a^{-n} = \frac{1}{a^n}$
• $a^{m/n} = \sqrt[n]{a^m}$
• \(\sin \theta = \frac{O}{H}, \cos \theta = \frac{A}{H}, \tan \theta = \frac{O}{A}\)
• $m = \frac{y_2 - y_1}{x_2 - x_1}$
• $t_n = t_1 + (n-1)d$

Key Terms

• Exponent: A number indicating how many times a base number is multiplied by itself.
• Radical: An expression involving a root, typically a square root or cube root.
• GCF: Greatest Common Factor; the largest number that divides into two or more numbers.
• LCM: Least Common Multiple; the smallest number that is a multiple of two or more numbers.
• SOH CAH TOA: Mnemonic for trigonometric ratios: Sine=Opposite/Hypotenuse, Cosine=Adjacent/Hypotenuse, Tangent=Opposite/Adjacent.
• Domain: The set of all possible input values (x-values) for a function.
• Range: The set of all possible output values (y-values) for a function.
• Slope: The measure of the steepness of a line, calculated as rise over run.
• Intercepts: Points where a line crosses the x-axis (x-intercept) or y-axis (y-intercept).
• Substitution Method: Solving a system of equations by expressing one variable in terms of another and substituting.
• Elimination Method: Solving a system of equations by adding or subtracting equations to eliminate one variable.

Real World Examples

• Calculating compound interest or population growth.: Exponent and radical rules.
• Determining dimensions of a box from its volume.: Factoring and polynomial expansion.
• Finding the height of a building using shadows or angles.: Right-angle trigonometry.
• Modeling costs based on fixed and variable components.: Linear functions and their equations.
• Solving problems involving two unknown quantities, like mixtures or rates.: Systems of linear equations.