Problem 88 - College Algebra

Question

Perform the indicated operation or operations. $$ \frac{(2 x-7)^{5}}{(2 x-7)^{3}} $$

Step-by-Step Solution

Answer

The simplified form of the expression is \((2x-7)^{2}\).

Step 1: Identify the Base and the Exponents

Notice that the base of both the numerator and the denominator of the fraction is \((2x-7)\). The exponent in the numerator is 5 and the exponent in the denominator is 3.

Step 2: Apply the Laws of Exponents

When dividing like bases, according to the laws of exponents, subtract the exponents. Thus, apply the rule: \(a^{m}/a^{n} = a^{m-n}\) to get \((2x-7)^{5-3}\).

Step 3: Simplify the Exponential Expression

Perform the subtraction in the exponent to simplify the expression. Thus, \((2x-7)^{5-3}\) simplifies to \((2x-7)^{2}\).

Key Concepts

Exponents

Exponents, sometimes referred to as powers, are a way to express how many times a number, known as the base, is multiplied by itself. For instance, in the expression \( (2x - 7)^5 \), "5" is the exponent, indicating that the base \((2x - 7)\) is multiplied by itself five times. Exponents follow specific rules called the laws of exponents, which greatly simplify calculations, especially when dealing with division and multiplication. These laws help us break down and manage complex algebraic expressions.One of the key laws is when you divide like bases: \( a^m/a^n = a^{m-n} \). This means you subtract the exponent of the denominator from the exponent of the numerator. This is a crucial step to master when simplifying expressions with exponents. Being comfortable with this concept allows you to tackle more complex algebraic questions with confidence.

Simplifying Expressions

Simplifying expressions involves using math rules to combine like terms and reduce an expression to its simplest form. It's about making expressions easier to understand and work with. When we simplify, we apply rules systematically.In the exercise \[ \frac{(2x-7)^5}{(2x-7)^3} \] we simplify by applying the exponent rule for division. The same base \((2x-7)\) is in both the numerator and the denominator. We used the rule \( a^m/a^n = a^{m-n} \), subtracting the smaller exponent from the larger one.In simple terms: - Identify the base and exponents- Use laws of exponents- Perform the calculationsAfter subtracting, \((2x-7)^{5-3}\) becomes \((2x-7)^2\), achieving the simplification. The key is understanding which rules to apply and practice will make these steps second nature.

Fractions

Fractions represent parts of a whole and are often expressed as \(\frac{a}{b}\), where \(a\) is the numerator and \(b\) is the denominator. In algebraic fractions like \(\frac{(2x-7)^5}{(2x-7)^3}\), we often use properties of numbers to simplify them by canceling common factors or applying the laws of exponents.When dealing with fractions involving exponents, identify common bases in both the numerator and the denominator. As in the exercise given, once the base and exponents are recognized, simplify by subtracting exponents if the bases match - an essential aspect of exponent rules for fractions.Moreover, simplifying fractions correctly can help avoid mistakes and make more complex problems clearer. It's a critical skill in math, enabling you to handle various algebraic expressions smoothly. Always ensure you apply the right rule and check your steps for accuracy in every fraction simplification task.