Rewriting algebraic expressions — factoring, expanding, combining rational expressions, and applying exponent rules to show two forms are equivalent.
82
Total questions
40
Easy
24
Medium
18
Hard
These questions ask which expression is equivalent to a given one: factoring quadratics and difference of squares, expanding products, simplifying rational expressions, and applying exponent and radical rules. No context, no story — pure algebraic manipulation, which makes them fast points once the rules are automatic.
Know the identities cold: x² − y² = (x+y)(x−y), (x+y)² = x² + 2xy + y², and the exponent laws including negative and fractional exponents (x^(a/b) is the b-th root of x^a). When manipulation gets messy, cheat numerically: plug x = 2 into the original and every answer choice — only the equivalent one matches.
Straight from the Grind1600 question bank — try each one before revealing the answer.
Correct answer: D
Choice D is correct. Applying the commutative property of multiplication, the expression becomes (4 · 2)(x³ · x⁵). Multiplying the coefficients gives 8, and for positive values of x, x³ · x⁵ = x3+5 = x⁸. Therefore, the expression equals 8x⁸. Choice B is incorrect and may result from multiplying the exponents. Choice C is incorrect and may result from adding the coefficients. Choice A is incorrect and may result from both errors.
Correct answer: B
Choice B is correct. One of the properties of radicals is ⁿ√(ab) = ⁿ√a · ⁿ√b. Thus, the given expression can be rewritten as ⁴√(a⁴) · ⁴√(b¹²). Simplifying by taking the fourth root of each part gives a¹ · b³, or ab³.
Choice A is incorrect and may be the result of ignoring the a⁴ term under the radical.
Choice C is incorrect and may result from dividing the exponent 12 by 3 instead of 4.
Choice D is incorrect and may result from dividing the exponent 12 by 3 instead of 4 and incorrectly simplifying.
Nonlinear Equations & Systems
Solving quadratic, radical, rational, and exponential equations, plus systems that mix a line with a curve — including discriminant reasoning.
Nonlinear Functions
Quadratic, exponential, and polynomial functions: vertex form, growth and decay, end behavior, and interpreting key features in context.
Function Notation
Evaluating and composing functions written as f(x), interpreting what f(a) = b means, and translating between notation, tables, and graphs.
82 Equivalent Expressions questions with step-by-step explanations, woven into a day-by-day study plan built for your test date.
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