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

Verified step by step guidance

1. Identify the given expression: \((2x - 7)^5 / (2x - 7)^3\). This is a division of exponential expressions with the same base \((2x - 7)\).
2. Recall the property of exponents: \(\frac{a^m}{a^n} = a^{m-n}\), where \(a\) is the base, and \(m\) and \(n\) are the exponents. Apply this property to simplify the expression.
3. Subtract the exponents: \(5 - 3\). This gives the new exponent for the base \((2x - 7)\).
4. Rewrite the simplified expression as \((2x - 7)^2\).
5. The final simplified expression is \((2x - 7)^2\). You can leave it in this form or expand it further if needed.

Key Concepts

Exponents and Power Rules

Understanding exponents is crucial in algebra, particularly the power rules which state that when dividing like bases, you subtract the exponents. For example, a^m / a^n = a^(m-n). This rule simplifies expressions involving powers and is essential for solving the given problem.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves reducing them to their simplest form. This process often includes combining like terms and applying exponent rules. In the context of the given question, simplifying (2x−7)^5/(2x−7)^3 requires applying the exponent subtraction rule to achieve a more manageable expression.

Polynomial Functions

Polynomial functions are expressions that involve variables raised to whole number exponents. The expression (2x−7) is a polynomial, and understanding its behavior, including how to manipulate it through operations like division, is fundamental in algebra. This knowledge helps in recognizing the structure of the expression and applying the correct operations.