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Chapter 0: Problem 3

Simplify the expression. \(\frac{w^{-4}}{v^{-2}}\)

Short Answer

Expert verified
The simplified expression is \(\frac{v^2}{w^4}\).

Step by step solution

01

Identify Negative Exponents

In the expression \(\frac{w^{-4}}{v^{-2}}\), there are negative exponents. Recall that \(a^{-n} = \frac{1}{a^n}\).
02

Rewrite Negative Exponents

Rewrite each negative exponent as a positive exponent: \(\frac{w^{-4}}{v^{-2}} = \frac{1}{w^4} \times v^2 = \frac{v^2}{w^4}\).
03

Final Simplified Expression

The expression has been simplified to \(\frac{v^2}{w^4}\). There are no further operations to perform.

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Key Concepts

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

Understanding Negative Exponents
Negative exponents might seem intimidating at first, but they follow a simple rule. When you see an expression like \(a^{-n}\), it means the reciprocal of \(a^n\). For example, \(x^{-3}\) is the same as \(\frac{1}{x^3}\). This is a fundamental concept in algebra that helps simplify expressions that involve both negative and positive exponents.

In our exercise, we had \(w^{-4}\) and \(v^{-2}\). We can rewrite these negative exponents as reciprocals:\(\frac{1}{w^4}\) and \(v^2\).
This step-by-step translation from negative to positive exponents is crucial for simplifying the overall rational expression.
Algebraic Simplification Techniques
Algebraic simplification is all about making an expression easier to work with. This often involves combining like terms, factoring, and reducing fractions.

In our exercise, we dealt with the expression \(\frac{w^{-4}}{v^{-2}}\). After recognizing the negative exponents, we rewrite them in a more manageable form:\(\frac{1}{w^4} \times v^2\). Now, multiplying the terms gives us \(\frac{v^2}{w^4}\). This process of rewriting and combining helps to clarify the expression, making it simpler and more understandable.
Simplification is a vital skill in algebra, allowing you to tackle more complex problems with ease.
Working with Rational Expressions
Rational expressions are fractions where the numerator and/or the denominator are polynomials. Simplifying these expressions is a key algebraic skill.

For our exercise, the rational expression was \(\frac{w^{-4}}{v^{-2}}\). By addressing the negative exponents first and rewriting them, we converted it to \(\frac{v^2}{w^4}\).
Understanding how to simplify rational expressions involves:
  • Identifying any negative exponents and converting them.
  • Combining and reducing terms where possible.
This process not only makes the expressions simpler but also prepares you for solving more advanced algebraic equations that you will encounter later on.
With practice, working with rational expressions becomes a straightforward and logical task.

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Most popular questions from this chapter

Multiply and simplify. $$ \left(4 u^{2}-5 v^{2}\right)\left(2 u^{2}+3 v^{2}\right) $$

Multiply and simplify. $$ \left(-10 c^{2} d^{5}\right)\left(\frac{1}{2} c d^{7}\right) $$

Suppose that \(x\) represents the larger of two consecutive odd integers. a. Write a polynomial that represents the smaller integer. b. Write a polynomial that represents the sum of the two integers. Then simplify. c. Write a polynomial that represents the product of the two integers. Then simplify. d. Write a polynomial that represents the difference of the squares of the two integers. Then simplify.

Perform the indicated operations and simplify. $$ (5 m-3)^{2} $$

Multiply and simplify. Assume that all variable expressions represent positive real numbers. $$ 3 \sqrt{6}(5 \sqrt{3}-4 \sqrt{2}-\sqrt{6}) $$
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