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Chapter 0: Problem 24

Simplify each exponential expression. $$x y^{-3}$$

Short Answer

Expert verified
The simplified form of the exponential expression \(x y^{-3}\) is \(\frac{x}{y^3}\).

Step by step solution

01

Identify the term to simplify

In the given expression \(x y^{-3}\), the term that needs simplification is \(y^{-3}\). This term represents y raised to the power of negative three.
02

Simplify the exponent

The operation \(y^{-3}\) can be rewritten as \(\frac{1}{y^3}\). That's because any number (except zero) to the power of a negative integer, can be written as one divided by that number raised to the positive of that integer.
03

Substitute the simplified term into the expression

Substitute \(\frac{1}{y^3}\) back into the original expression replacing \(y^{-3}\). Now, \(x y^{-3}\) becomes \(x \times \frac{1}{y^3}\) which finally simplifies to \(\frac{x}{y^3}\).

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Key Concepts

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

Negative Exponents
Negative exponents can often seem confusing at first, but they're pretty simple once you get the hang of them. A negative exponent indicates that instead of multiplying, you divide. It's a way of denoting reciprocals using exponential notation. For example, when you see something like the expression \(y^{-3}\), it doesn't mean miss out on the fun; it means to take the reciprocal. So, \(y^{-3}\) is equivalent to \(\frac{1}{y^3}\). This rule applies universally to any base. Here's a friendly reminder:
  • If you have a negative exponent in a term, you can move it to either the numerator or denominator of a fraction, just flip the exponent's sign when you do!
  • For example, \(a^{-n} = \frac{1}{a^n}\).
With this simple transformation, negative exponents become much less intimidating!
Simplifying Expressions
Reducing expressions to their simplest form is a foundational math skill. It requires us to transform expressions into something more straightforward without changing their value. In the case of exponential expressions, we often simplify them to make calculations easier.
For the expression \(xy^{-3}\), start by identifying parts that can be simplified. Here, it's the term with the negative exponent \(y^{-3}\). By converting \(y^{-3}\) into its reciprocal form \(\frac{1}{y^3}\), we simplify the expression:
  • Replace \(y^{-3}\) with \(\frac{1}{y^3}\).
  • The expression becomes \(x \times \frac{1}{y^3}\).
  • Which simplifies further to \(\frac{x}{y^3}\).
The goal of simplifying is to make expressions easier to read and work with, paving the way to solve more complex problems.
Exponential Rules
Exponential rules are like trusty roadmaps guiding you through the land of algebra. They help you manipulate and simplify expressions that contain exponents, whether positive or negative. Here are some essential rules to keep in mind:
  • Product of powers rule: \(a^m \times a^n = a^{m+n}\).
  • Quotient of powers rule: \(\frac{a^m}{a^n} = a^{m-n}\) where \(a eq 0\).
  • Power of a power rule: \((a^m)^n = a^{m \times n}\).
  • Power of a product rule: \((ab)^n = a^n \times b^n\).
  • Negative exponent rule: \(a^{-n} = \frac{1}{a^n}\).
These rules aren't just theoretical; they provide the keys to unlock simpler forms of complex expressions. With practice, these become second nature, aiding in quick and efficient problem-solving.

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Most popular questions from this chapter

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. An elevator at a construction site has a maximum capacity of 3000 pounds. If the elevator operator weighs 245 pounds and each cement bag weighs 95 pounds, how many bags of cement can be safely lifted on the elevator in one trip?

Describe how to solve an absolute value inequality involving the symbol \(>.\) Give an example.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I prefer interval notation over set-builder notation because it takes less space to write solution sets.

a. A mathematics professor recently purchased a birthday cake for her son with the inscription $$ \text { Happy }\left(2^{\frac{5}{2}} \cdot 2^{\frac{3}{4}} \div 2^{\frac{1}{4}}\right) \text { th Birthday. } $$ How old is the son?b. The birthday boy, excited by the inscription on the cake, tried to wolf down the whole thing. Professor Mom, concerned about the possible metamorphosis of her son into a blimp, exclaimed, "Hold on! It is your birthday, so why not take \(\frac{8^{-\frac{4}{3}}+2^{-2}}{16^{-\frac{3}{4}}+2^{-1}}\) of the cake? I'll eat half of what's left over." How much of the cake did the professor eat?

In Exercises \(133-136\), determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\frac{\sqrt{20}}{8}=\frac{\sqrt{10}}{4}$$
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