Podcast Weaknesses PDF

## Quantitative Concept #4: Exponents and Roots - If math isn't your thing, you can figure out questions with exponents. - Exponents are a way to multiply a number by itself a certain number of times. - The expression 5⁴ means multiply four 5's together. - 2³ means multiply three 2's together: 2³ =...

## Quantitative Concept #4: Exponents and Roots - If math isn't your thing, you can figure out questions with exponents. - Exponents are a way to multiply a number by itself a certain number of times. - The expression 5⁴ means multiply four 5's together. - 2³ means multiply three 2's together: 2³ = 8. - The base is the number that is multiplied by itself. - The exponent is the number of times the base is multiplied by itself. - The power is the result of exponentiation. ### The Lowdown on Exponents - **Fundamentally**, an exponent is how many times you multiply a number. - **Exponentiation** distributes over multiplication and division. - **Rearranging factors** in multiplication does not change the result. - Exponentiation does **not** distribute over addition. - (a + b)ⁿ ≠ aⁿ + bⁿ ### Multiplying Powers - When multiplying two powers of the same base, **add the exponents**: - (xᵐ) x (xⁿ) = xᵐ⁺ⁿ ### Dividing Powers - When dividing two powers of the same base, **subtract the exponents**: - (xᵐ) / (xⁿ) = xᵐ⁻ⁿ ### An Exponent of 0 - **Anything to the power of 0 equals 1**: x⁰ = 1 ### Negative Exponents - x⁻ⁿ = 1/xⁿ ## The Lowdown on Roots - **Square root** is the inverse of squaring: √36 = 6. - It's easy to find the square root of a perfect square. - All other square roots are decimals. - **Cube root** is the inverse of cubing: ³√8 = 2. - The cube root of a positive number is positive. - The cube root of a negative number is negative. ## The Symbol: Positive or Negative - The symbol √ represents the **principal square root**, meaning only the positive value. - √16 = +4, even though (-4)² = 16. - We can take the square root of 0: √0 = 0. - We **cannot** take the square root of a negative (e.g., √-1) ## Cubes and Cube Roots - **Cubing** a number means multiplying it by itself three times: 2³ = 8. - The cube root of a perfect cube is easy to find. - Cubes and cube roots with negatives gets interesting: - (-2)³ = -8. - The cube root of a negative number is negative. ## Quantitative Concept #3: Algebra - Algebra represents unknown numbers as letters. - The GRE algebra tends to be harder than basic equations. - **Suggestions:** - Simplify the given equation by dividing or multiplying both sides. - Clear any fractions by multiplying both sides by the denominator. - Don't ignore the wording of the problem. ### Algebraic Concepts: The Lowdown - **Multiplying both sides by the unknown:** If x is guaranteed to be non-zero you can multiply both sides of the equation by x. - **Adding or subtracting multiples of the unknown:** You can add or subtract multiples of the unknown (e.g., 3x). - **Dividing by the unknown:** You **cannot** divide by an unknown variable unless you are completely certain that it cannot be zero. ### Working with Quadratics - A **monomial** is a product of numbers and variables. - A **polynomial** is the sum of terms. - A **binomial** is a polynomial with two terms. - A **trinomial** is a polynomial with three terms. - **FOIL** (First, Outer, Inner, Last) is a method for multiplying two binomials: - (3x + 5) * (2x - 7) = 6x² - 21x + 10x - 35 = 6x² - 11x - 35 - **Factoring** a quadratic is converting it into the product of two binomials. - **To factor a quadratic:** - Find two numbers whose product equals the constant term and whose sum equals the coefficient of the middle term. ## Quantitative Concept #2: Inequalities - An inequality is a mathematical statement that uses one of the following signs: - <, >, ≤, ≥ ### Rules of Inequalities - **Adding and subtracting equal quantities:** You can add or subtract the same quantity to both sides of an inequality. - **Multiplying and dividing by a positive number:** You can multiply or divide both sides of an inequality by a positive number. - **Multiplying and dividing by a negative number:** Multiplying or dividing both sides of an inequality by a negative number reverses the inequality (e.g., if x > 5, then -x < -5). ## Solving Systems of Equations - A system of equations is a set of two or more equations with the same unknowns. ### Solving Systems: Substitution - **Steps:** 1. Solve one equation for one unknown. 2. Substitute the expression for that unknown into the other equation. 3. Solve the resulting equation for the remaining unknown. 4. Substitute the value of that unknown back into one of the original equations to solve for the first unknown. ### Solving Systems: Elimination - **Steps:** 1. Multiply one or both equations by numbers so that the coefficients of one variable in the two equations have equal absolute values and opposite signs. 2. Add or subtract the equations, eliminating one variable. 3. Solve the resulting equation for the remaining unknown. 4. Substitute the value of that unknown back into one of the original equations to solve for the first unknown. ### Essential Algebra Formulas - **Difference of Two Squares:** A² - B² = (A + B)(A - B) - **Square of a Sum:** (A + B)² = A² + 2AB + B² - **Square of a Difference:** (A - B)² = A² - 2AB + B² ## Two Equations with Two Variables - A single equation with two variables can be graphed as a line. - To solve for two variables, you need two equations (which will be graphed as two lines). - The intersection point of the two lines represents the solution to the system of equations. ### Solving Systems: Substitution - The substitution method relies on the idea that two equal quantities can be interchanged. ### Solving Systems: Elimination - The elimination method involves manipulating the equations so that one variable cancels out when the equations are added or subtracted. - You can add or subtract the same quantity to both sides of an equation. - You can multiply or divide both sides of an equation by the same number. ## Two Equations with Two Variables: Practice Problems - **Identify the best approach (substitution or elimination) based on the coefficients.** - **Practice solving the equation.** **Note: This is a long document. It may be helpful to break it down into multiple files for easier reading.**