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Chapter 5: Problem 101

Simplify. Do not use negative exponents in the answer. \(x^{-3} \cdot x^{-3}\)

Short Answer

Expert verified
\( \frac{1}{x^{6}} \)

Step by step solution

01

Apply the Product of Powers Property

When multiplying powers with the same base, you can add the exponents. The expression is given as: \[ x^{-3} imes x^{-3} \] Using the product of powers property: \[ x^{a} imes x^{b} = x^{a+b} \] The expression becomes: \[ x^{-3 + (-3)} = x^{-6} \]
02

Convert Negative Exponent to Positive Exponent

To express the power with a positive exponent, use the negative exponent rule: \[ x^{-a} = \frac{1}{x^{a}} \] For our problem: \[ x^{-6} = \frac{1}{x^{6}} \]
03

Final Simplification

Thus, the simplified expression without negative exponents is \[ \frac{1}{x^{6}} \]

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Key Concepts

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

Product of Powers Property
The Product of Powers Property is a rule that helps simplify expressions when multiplying two or more numbers with the same base. It states that when you multiply powers with the same base, you simply add the exponents together. This rule can be expressed as:
  • \[ x^a \times x^b = x^{a+b} \]
For example, if you have \(x^{-3} \times x^{-3}\), using this property means you add the exponents \(-3 + (-3)\) which results in \(x^{-6}\). Being able to use the Product of Powers Property simplifies the process of handling expressions with multiple powers. This rule is vital when it comes to working with exponential expressions because it allows you to combine and reduce expressions into simpler forms.
Negative Exponent Rule
The Negative Exponent Rule is crucial to understanding how to work with powers that have negative exponents. A negative exponent indicates that the base is on the "wrong" side of a fraction, and getting it to the "correct" side requires flipping the base. This rule can be summarized as:
  • \[ x^{-a} = \frac{1}{x^a} \]
When you encounter a situation like \(x^{-6}\), this rule tells you to rewrite it as \(\frac{1}{x^6}\). The Negative Exponent Rule allows you to transform expressions with negative powers into a more standardized, up-front form without any negatives involved. It's especially helpful for final answers, where the convention is to express the results using positive exponents only.
Simplifying Expressions with Exponents
Simplifying expressions with exponents involves applying key rules and properties to make an expression as concise as possible. When simplifying, the goal is to reduce the expression using as few terms as possible while maintaining mathematical accuracy. Here's a basic plan for simplifying expressions with exponents:
  • Apply the Product of Powers Property when multiplying expressions with the same base, allowing you to add exponents together.
  • Use the Negative Exponent Rule to eliminate negative exponents in the final answer, converting them into a fraction with positive exponents.
  • Review the expression to ensure it's fully simplified, including ensuring all exponents are positive and no further reduction is possible.
In the case of the expression \(x^{-3} \times x^{-3}\), after applying the product property to get \(x^{-6}\), and the negative exponent rule, we simplify it to \(\frac{1}{x^6}\). Understanding these steps ensures that students can handle similar problems with ease and confidence.

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