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Chapter 5: Problem 40

\(9 u^{-5} v^{2}\)

Short Answer

Expert verified
\(9 v^{2} / u^{5}\)

Step by step solution

01

Identify components of the expression

Recognize the parts of the expression: the numerical coefficient 9, variable 'u' with power -5, and 'v' with power 2.
02

Rewrite 'u' with a negative exponent

Write 'u' as a fraction in the denominator of the main fraction, since it has a negative exponent. So, you should transform \(u^{-5}\) into \(1 / u^{5}\).
03

Combine components

Gather all of the components together: \(9 v^{2} / u^{5}\). This is the simplified version of the expression.

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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 a bit challenging at first, but once you understand the rule behind them, they become much easier to manage. An exponent indicates how many times a number, or a variable, is multiplied by itself. When that exponent is negative, it signifies reciprocity.
For example, if you have a term like \( u^{-5} \), the negative exponent suggests that \( u \) should actually be in the denominator of a fraction. This is because \( u^{-5} = \frac{1}{u^5} \). It is simply the process of flipping the position from the numerator to the denominator.
  • This makes negative exponents helpful for expressing fractions without having to carry fractions around initially.
  • The key to dealing with them is to remember that they reverse the position in a fraction, making it easy to switch between numerator and denominator.
Introduction to Algebraic Expressions
Algebraic expressions are a fundamental part of algebra and arithmetic. They consist of numbers, variables, and operations combined. For example, in the expression \(9 u^{-5} v^2\), the numbers and letters together describe a math idea or a set of math operations.
For this expression:
  • "9" is the numerical coefficient, showing how many times the variable part of the expression is multiplied.
  • "u" and "v" are variables, meaning they can represent different values.
  • Exponents above variables indicate repeated multiplication.
Recognizing these components helps in manipulating these expressions to solve equations, analyze relationships, or simplify expressions.
Simplifying Expressions Made Easy
Simplifying expressions means to rewrite them in a simpler or reduced form without changing their value. It often involves working with exponents, coefficients, and variables by using algebraic rules. With the expression \(9 u^{-5} v^2\), simplifying involves rearranging terms to make them easier to understand and work with.
Here鈥檚 how you do it:
  • Identify parts of the expression 鈥� recognize coefficients, find variables with exponents.
  • Rewrite negative exponents - convert them into fractions to make them positive.
  • Group and rearrange - gather all rewritten terms into a single fraction to clarify the expression鈥檚 simplest form.
In our example, by rewriting \(u^{-5}\) as \(\frac{1}{u^5}\), we start simplifying the expression step-by-step to end up with \(\frac{9v^2}{u^5}\). This approach helps in making complex equations or expressions more manageable.

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