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Chapter 2: Problem 125

Simplify the following problems. $$ \frac{u^{w}}{u^{k}} $$

Short Answer

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Question: Simplify the expression \(\frac{u^{w}}{u^{k}}\). Answer: \(u^{w-k}\)

Step by step solution

01

Identify terms and exponents

We are given the expression: \(\frac{u^{w}}{u^{k}}\). The base for both terms is "u" and the exponents are "w" and "k".
02

Apply the quotient of powers property

The quotient of powers property states that when dividing two exponential expressions with the same base, you subtract the exponents: \(a^{m} 梅 a^{n} = a^{m-n}\). Applying the property to our expression, we get \(u^{w-k}\).
03

Write the simplified expression in the given form

The simplified expression is: \(u^{w-k}\).

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Key Concepts

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

Quotient of Powers
The quotient of powers is an essential property in mathematics that helps simplify expressions involving exponents. When we are dealing with two exponential terms that have the same base, and we are dividing one by the other, we can use this property.

Simply put, the quotient of powers property tells us that we should subtract the exponents. For example, if you have an expression like \( a^{m} \) divided by \( a^{n} \), it simplifies to \( a^{m-n} \). Here, "a" represents the base, and "m" and "n" are the exponents. This rule only applies if the bases of the exponents are the same.
  • This property makes algebraic manipulations much more straightforward because instead of working with complex fractions, we can focus on the difference of the exponents.
  • Understanding this property is crucial for anyone looking to master the basics of algebra and handle expressions involving exponents effectively.
Simplifying Expressions
Simplifying expressions is a fundamental skill in algebra that involves reducing an expression to its simplest form. This improves clarity and often makes further calculations easier. To simplify an expression like \( \frac{u^{w}}{u^{k}} \), we leverage properties of exponents.
  • First, identify if there are common bases in both the numerator and the denominator.
  • Then, use the quotient of powers property to simplify the expression by subtracting the exponent in the denominator from the exponent in the numerator.
Take the expression \( \frac{u^{w}}{u^{k}} \) as an example. The base "u" is common, and thus, the expression simplifies to \( u^{w-k} \).

Simplification not only helps with solving math problems but also aids in better understanding and communicating mathematical ideas effectively.
Algebraic Manipulation
Algebraic manipulation refers to the process of rearranging and simplifying algebraic expressions for various mathematical purposes. It involves applying rules and properties to transform an expression into a desired form.

When dealing with expressions involving exponents, algebraic manipulation can include:
  • Applying the quotient of powers property to combine or simplify expressions.
  • Recognizing patterns or common terms that can aid in the simplification process.
  • Ensuring that expressions are correctly rewritten to facilitate easier calculations or comparisons.
In the exercise \( \frac{u^{w}}{u^{k}} \), algebraic manipulation is employed by utilizing the known property of exponents to rewrite the expression as \( u^{w-k} \).

Such manipulation is key in higher mathematics and allows you to solve equations, evaluate expressions more efficiently, and understand mathematical relationships deeply.

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