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Chapter 8: Problem 88

Simplify each expression. $$\left(2 x^{3}\right)^{-1}$$

Short Answer

Expert verified
1/(2x^3)

Step by step solution

01

Understand the Negative Exponent Rule

The negative exponent rule states that \(a^{-n} = \frac{1}{a^n}\). In this context, \[ \left(2x^3\right)^{-1} = \frac{1}{\left(2x^3\right)} \] Use this rule to transform the given expression.
02

Apply the Rule to the Expression

Apply the negative exponent rule to the given expression: \[ \left(2x^3\right)^{-1} = \frac{1}{2x^3} \] This step simplifies the expression using the rule from the previous step.

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Key Concepts

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

Simplifying Expressions
When we talk about simplifying expressions, we mean making them easier to read, understand, and work with. In algebra, simplifying an expression often involves performing operations and applying rules to rewrite the expression in the most straightforward form.

For example, look at the expression \( \left(2x^3\right)^{-1} \). To simplify this, we use the concept of negative exponents.

  • First, identify parts of the expression that can be simplified.
  • Then, apply algebraic rules such as the negative exponent rule to simplify the expression.
Simplifying expressions is crucial because it makes solving algebraic equations and understanding their components much easier.
Exponents
Exponents are a way to represent repeated multiplication of the same number. For instance, \(x^3\) means \(x \times x \times x\). Here are some key points about exponents that help in understanding how to work with them:

  • Basic Rule: \ a^n \ means \ a \ times itself \ n \ times.
  • Product Rule: When you multiply like bases, you add the exponents: \( a^m \times a^n = a^{m+n} \).
  • Quotient Rule: When you divide like bases, you subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \).
  • Negative Exponent Rule: An exponent of \ -n \ means taking the reciprocal of the base raised to the positive \ n \: \( a^{-n} = \frac{1}{a^n} \).
Understanding these basic rules makes working with exponents much easier and sets a strong foundation for more complex algebraic operations.
Algebra Rules
Algebra includes a variety of rules that help us solve equations and understand expressions. Here are some commonly used algebraic rules:

  • Commutative Rule: The order in which you add or multiply numbers does not change the result: \( a + b = b + a \) and \( a \times b = b \times a \).
  • Associative Rule: The way you group numbers when adding or multiplying does not change the result: \( (a + b) + c = a + (b + c) \) and \( (a \times b) \times c = a \times (b \times c) \).
  • Distributive Rule: Distributes multiplication over addition: \( a \times (b + c) = a \times b + a \times c \).
  • Negative Exponent Rule: This states that \( a^{-n} = \frac{1}{a^n} \); it helps transform negative exponents into positive ones, making expressions easier to handle.
Applying these algebra rules properly allows you to simplify and solve various algebraic problems effectively.

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