Problem 89 - Algebra and Trigonometry

Question

In Exercises 83–90, perform the indicated operation or operations. $$ \frac{(2 x-7)^{5}}{(2 x-7)^{3}} $$

Step-by-Step Solution

Answer

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

Step 1: Identify the base

The values \(2x-7\) to the exponent in the numerator and denominator form the base of the power.

Step 2: Apply the rule of exponents

Since the base is the same in both numerator and denominator, we can simplify the expression by subtracting the exponent of the denominator from the exponent of the numerator. In this case, raise the base \(2x-7\) to the power \((5-3)\), which simplifies the expression to \( (2x-7)^{2} \).

Key Concepts

Rule of Exponents

When simplifying rational expressions involving exponents, understanding the rules of exponents is crucial. Essentially, these rules provide us with shortcuts to simplify expressions without expanding them.

Consider the expression \(\frac{(2x-7)^{5}}{(2x-7)^{3}}\). To simplify such an expression, we look at two critical components: the base \(2x-7\) and the respective exponents 5 and 3. According to the rules of exponents, when we divide like bases, we subtract the exponent in the denominator from the exponent in the numerator.

Thus, applying this rule simplifies the expression to \( (2x-7)^{5-3} \) which further reduces to \( (2x-7)^{2} \) without the need for expansion.

• Identifying the base and keeping it consistent is the first step.
• Then, subtract the lower exponent from the higher one when dividing powers with the same base.

Being familiar with this rule allows for quick and easy simplification, aiding in a more efficient problem-solving process.

Algebraic Operations

Algebraic operations encompass the fundamental procedures used to manipulate algebraic expressions, which include addition, subtraction, multiplication, division, and exponentiation. Simplifying rational expressions often requires a combination of these operations.

In our exercise, the division of two expressions with exponents mainly involves the operation of exponentiation and its properties. However, it's important to understand that before one can apply the properties of algebraic operations such as exponentiation, terms must be in like form.

This means the bases in our example, \(2x-7\), are the same, allowing the exponents' rule to be applied directly. When working with algebraic expressions, always look out for like terms and apply operations accurately to simplify efficiently.

• Recognize the operation required (in this case, division).
• Ensure terms are in like form before proceeding.
• Apply the algebraic operations systematically.

Power of a Power Property

Another important property within the realm of exponents is the power of a power property. This comes into play when we are dealing with an expression that already has an exponent and needs to be raised to another exponent.

The power of a power property states that to raise a power to a power, you multiply the exponents. For example, if we had the expression \( (2x-7)^{2} \) and wanted to raise this entire expression to the 3rd power, we would multiply the exponents: \( (2x-7)^{2*3} \) which results in \( (2x-7)^{6} \).

In the context of our original problem, this property did not need to be directly applied because we were simplifying by subtracting exponents. However, understanding this property is essential when dealing with more complex exponentiation within algebraic expressions.

• Multiplication of exponents comes into effect when an exponentiated term is raised to another power.
• This property connects multiplication and exponentiation in the context of algebraic operations.