In Exercises 83–90, perform the indicated operation or operations. (2x−7)^5/(2x−7)^3

Verified step by step guidance

1. Step 1: Recognize that this problem involves the properties of exponents. Specifically, when you divide two expressions with the same base, you subtract the exponents.
2. Step 2: In this case, the base is (2x-7) and the exponents are 5 and 3.
3. Step 3: Subtract the exponent of the denominator from the exponent of the numerator. This gives you (2x-7)^(5-3).
4. Step 4: Simplify the exponent to get (2x-7)^2.
5. Step 5: If necessary, expand the expression to get 4x^2 - 28x + 49.

Key Concepts

Exponents and Power Rules

Exponents represent repeated multiplication of a base number. The power rules, particularly the quotient rule, state that when dividing two expressions with the same base, you subtract the exponents. For example, a^m / a^n = a^(m-n). This rule is essential for simplifying expressions involving exponents.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves reducing them to their simplest form by combining like terms and applying mathematical operations. This process often includes factoring, distributing, and using exponent rules to make expressions easier to work with. Mastery of simplification is crucial for solving equations and performing operations.

Polynomial Functions

Polynomial functions are expressions that consist of variables raised to non-negative integer powers, combined using addition, subtraction, and multiplication. In the given expression, (2x−7) is a polynomial, and understanding its behavior and properties is important for performing operations like division. Recognizing the structure of polynomials aids in effective manipulation and simplification.