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Chapter 11: Problem 8

Simplify. $$\frac{(2 x-1)^{5}}{(2 x-1)^{4}}$$

Short Answer

Expert verified
\((2x - 1)\)

Step by step solution

01

Identify Base and Powers

The base for both terms is (2x - 1), with respective powers of 5 and 4.
02

Apply the Subtraction Rule for Exponents

When dividing terms with the same base, subtract the exponent of the divisor from the exponent of the dividend. Meaning \((2x - 1)^{5} / (2x - 1)^{4}\) equals \((2x - 1)^{5-4}\).
03

Simplify the Exponent

Finally, simplify the exponent 5 - 4 = 1. This means the whole expression simplifies to \((2x - 1)^{1}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponent Rules
Exponent rules are like special guidelines that help us work with numbers raised to powers. These rules simplify expressions and are crucial for tackling complex mathematical problems with ease. One of the most important rules is the "Power of a Quotient" rule, which states that when dividing expressions with the same base, you subtract the exponents.
For example:
  • When dividing \(a^m\) by \(a^n\), the result is \(a^{m-n}\).
  • This means if you have \(x^5 / x^3\), it simplifies to \(x^{5-3} = x^2\).
Understanding exponent rules allows you to manipulate and simplify expressions quickly, which is extremely helpful in both algebra and calculus tasks. When you recognize common bases in terms, applying the correct exponent rule can lead to much simpler solutions.
Base and Powers
In mathematics, when we talk about bases and powers, we're referring to numbers involved in exponential expressions. Let's break it down:
  • The "base" is the number or expression being multiplied by itself.
  • The "power" or "exponent" indicates how many times the base is used as a factor.
For instance, in the expression \(a^m\), "a" is the base and "m" is the exponent, which means that "a" is successfully multiplied by itself "m" times. In the given exercise, both the numerator and the denominator have the same base, \(2x-1\). So, you only need to focus on changing the powers according to the operation needed鈥攊n this case, division.
Recognizing the base is critical, as it helps you determine if and how exponent rules can be applied.
Exponent Subtraction
Exponent subtraction is used when dividing numbers with the same base. It is a straightforward operation that can simplify expressions significantly. Let's see how it works:
When you divide two expressions with the same base, you write the expression again with the subtracted value of the exponents.
  • If you have \(a^m / a^n\), you rewrite it as \(a^{m-n}\).
In our case, the problem involved dividing \((2x - 1)^5\) by \((2x - 1)^4\). By applying exponent subtraction, \(5 - 4 = 1\), you reduce the expression to \((2x - 1)^1\). This reduces complex problems into simpler forms, making calculations more manageable.
Whenever you identify like bases stacked with powers, exponent subtraction is the tool to simplify them easily.

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Most popular questions from this chapter

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