Algebra simplification
48 flashcards covering Algebra simplification for the GMAT Quantitative section.
Algebra simplification is the process of rewriting complex algebraic expressions into simpler, more manageable forms. This involves techniques like combining like terms, factoring, expanding, or canceling common factors. For instance, simplifying an expression such as 2x + 3x - 4x to just x makes equations easier to solve and helps reveal underlying patterns. Mastering this skill is fundamental because it builds the foundation for more advanced math problems, allowing you to work efficiently and avoid errors in calculations.
On the GMAT Quantitative section, algebra simplification frequently appears in problem-solving and data sufficiency questions, where you must manipulate expressions to evaluate options or derive answers. Common traps include mishandling negative signs, exponents, or fractions, which can lead to incorrect simplifications and wasted time. Focus on accuracy with basic rules, practicing under timed conditions to spot these pitfalls quickly and improve your overall performance.
Always double-check your work by plugging in numbers to verify simplifications.
Terms (48)
- 01
Like terms
Like terms are terms in an algebraic expression that have the same variables raised to the same powers, and they can be combined by adding or subtracting their coefficients to simplify the expression.
- 02
Distributive property
The distributive property states that multiplying a number by a sum inside parentheses means multiplying the number by each term inside, such as a(b + c) = ab + ac, which is used to expand or simplify expressions.
- 03
Combining like terms
Combining like terms involves adding or subtracting the coefficients of terms with identical variables and exponents, such as 3x + 2x = 5x, to simplify an algebraic expression.
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Factoring out the greatest common factor
Factoring out the greatest common factor means dividing each term in an expression by the largest factor that divides them all evenly, then writing it outside parentheses, such as 4x + 6 = 2(2x + 3).
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Difference of squares
The difference of squares is a factoring formula for an expression like a^2 - b^2, which factors into (a - b)(a + b), used to simplify or solve quadratic equations.
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Perfect square trinomial
A perfect square trinomial is an expression like a^2 + 2ab + b^2, which factors into (a + b)^2, or a^2 - 2ab + b^2 into (a - b)^2, helping in simplifying quadratics.
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Sum of cubes
The sum of cubes formula is a^3 + b^3 = (a + b)(a^2 - ab + b^2), used to factor and simplify cubic expressions in algebra problems.
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Difference of cubes
The difference of cubes formula is a^3 - b^3 = (a - b)(a^2 + ab + b^2), applied to factor and simplify cubic algebraic expressions.
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Factoring trinomials
Factoring trinomials involves breaking down a quadratic like ax^2 + bx + c into (dx + e)(fx + g), where the factors multiply to c and add to b, to simplify or solve equations.
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Exponent product rule
The exponent product rule states that when multiplying powers with the same base, add the exponents, such as a^m a^n = a^(m+n), to simplify expressions with exponents.
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Exponent quotient rule
The exponent quotient rule says that when dividing powers with the same base, subtract the exponents, like a^m / a^n = a^(m-n), for simplifying fractional exponents.
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Exponent power rule
The exponent power rule means that when raising a power to another power, multiply the exponents, such as (a^m)^n = a^(mn), to simplify complex exponents.
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Zero exponent rule
The zero exponent rule states that any non-zero number raised to the power of zero is 1, such as a^0 = 1, which simplifies expressions involving exponents.
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Negative exponent rule
The negative exponent rule indicates that a negative exponent means taking the reciprocal of the base, like a^(-n) = 1/a^n, used to simplify expressions with negative powers.
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Simplifying fractional exponents
Simplifying fractional exponents involves rewriting them as roots and powers, such as a^(1/2) = √a, and then simplifying based on exponent rules.
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Simplifying radicals
Simplifying radicals means reducing the expression under the root by factoring out perfect squares, such as √(16x^2) = 4x, to make it easier to work with.
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Adding radicals
Adding radicals requires combining like terms under the same root, such as √8 + √2 = 2√2 + √2 = 3√2, but only if the radicands are the same.
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Multiplying radicals
Multiplying radicals involves multiplying the radicands and the coefficients separately, like √a √b = √(ab), and then simplifying the result.
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Rationalizing the denominator
Rationalizing the denominator means eliminating radicals or complex numbers from the bottom of a fraction by multiplying numerator and denominator by the conjugate, such as (1/√2) (√2/√2) = √2/2.
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Simplifying algebraic fractions
Simplifying algebraic fractions involves factoring the numerator and denominator and canceling common factors, such as (2x + 4)/(x + 2) = 2(x + 2)/(x + 2) = 2, for x ≠ -2.
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Adding algebraic fractions
Adding algebraic fractions requires a common denominator, then combining numerators, such as (1/x) + (1/y) = (y + x)/(x y), and simplifying the result.
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Multiplying algebraic fractions
Multiplying algebraic fractions means multiplying numerators together and denominators together, then simplifying, like (a/b) (c/d) = (a c)/(b d).
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Dividing algebraic fractions
Dividing algebraic fractions involves multiplying by the reciprocal of the divisor, such as (a/b) ÷ (c/d) = (a/b) (d/c) = (a d)/(b c), and then simplifying.
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Simplifying complex fractions
Simplifying complex fractions means multiplying numerator and denominator by the least common denominator to eliminate inner fractions, such as (1 + 1/2)/(1 - 1/2) = (3/2)/(1/2) = 3.
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Solving linear equations
Solving linear equations involves isolating the variable by performing inverse operations, such as adding, subtracting, multiplying, or dividing both sides, like 2x + 3 = 7 simplifies to x = 2.
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Equations with variables on both sides
Equations with variables on both sides require moving all variable terms to one side and constants to the other, such as 2x + 3 = x + 5 simplifies to x = 2.
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Quadratic equations by factoring
Solving quadratic equations by factoring means setting the equation to zero, factoring into binomials, and using the zero product rule, like x^2 - 4 = 0 factors to (x-2)(x+2)=0.
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Quadratic formula
The quadratic formula is x = [-b ± √(b^2 - 4ac)] / (2a) for an equation ax^2 + bx + c = 0, used to find roots when factoring is not straightforward.
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Completing the square
Completing the square transforms a quadratic equation into vertex form by adding and subtracting a constant, such as x^2 + 6x = 9 becomes (x+3)^2 - 9 = 0.
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Absolute value equations
Absolute value equations involve isolating the absolute value and considering both positive and negative cases, like |x - 3| = 2 means x - 3 = 2 or x - 3 = -2.
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Inequality basics
Basic inequalities are solved like equations but reverse the sign when multiplying or dividing by a negative number, such as 2x > 4 simplifies to x > 2.
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Multiplying inequalities by negative numbers
When multiplying or dividing an inequality by a negative number, reverse the inequality sign, such as -2x > 4 simplifies to x < -2.
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Compound inequalities
Compound inequalities combine two inequalities, like a < x < b, and are solved by isolating x while maintaining the order, such as 2 < 3x + 1 < 7 simplifies to 1/3 < x < 2.
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Absolute value inequalities
Absolute value inequalities like |x| < a mean -a < x < a, and |x| > a means x < -a or x > a, solved by considering the definition of absolute value.
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Domain of a function
The domain of a function is the set of all possible input values for which the function is defined, such as excluding values that make denominators zero in rational functions.
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Range of a function
The range of a function is the set of all possible output values, determined after simplifying or analyzing the function's behavior, like for f(x) = x^2, the range is y ≥ 0.
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Linear functions
Linear functions are of the form f(x) = mx + b, and simplifying them involves solving for x or y, or graphing to understand their straight-line behavior.
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Quadratic functions
Quadratic functions are of the form f(x) = ax^2 + bx + c, and simplifying them often includes finding the vertex or roots to analyze their parabolic shape.
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Common sign errors in algebra
Common sign errors occur when distributing negative signs or factoring, such as forgetting to change signs inside parentheses, like -(x - 2) = -x + 2.
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Order of operations in algebra
The order of operations, or PEMDAS, dictates that parentheses, exponents, multiplication/division, and addition/subtraction are performed in that order to simplify expressions accurately.
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Substituting values into expressions
Substituting values means replacing variables with numbers in an expression and simplifying, such as plugging x=2 into 3x + 1 gives 3(2) + 1 = 7.
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Expanding binomials
Expanding binomials involves using the distributive property or FOIL for two terms, like (x + 2)(x + 3) = x^2 + 5x + 6, to simplify or prepare for further factoring.
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Simplifying expressions with parentheses
Simplifying expressions with parentheses requires distributing any multipliers and then combining like terms, such as 2(x + 3) + 4 = 2x + 6 + 4 = 2x + 10.
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Equations with fractions
Equations with fractions are simplified by multiplying through by the least common denominator to eliminate fractions, like (1/2)x + 3 = 5 becomes x + 6 = 10.
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Simplifying ratios
Simplifying ratios involves dividing both parts by their greatest common factor, such as 4:6 simplifies to 2:3, often in word problems involving proportions.
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Percent change in algebra
Percent change is calculated as [(new value - original value) / original value] 100%, and simplifying this expression helps in problems involving increases or decreases.
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Work-rate problems
In work-rate problems, rates are added or combined as fractions of work done per unit time, such as two people working together, and simplified to find total time.
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Mixture problems
Mixture problems involve setting up equations based on weighted averages, like mixing solutions of different concentrations, and simplifying to find the desired mixture.