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Chapter 5: Problem 58

Simplify each expression. Write answers using positive exponents.. $$ -w^{-2} $$

Short Answer

Expert verified
-\(\frac{1}{w^2}\) is the simplified form using positive exponents.

Step by step solution

01

Identify the Expression

The expression given is \(-w^{-2}\). This expression has a negative exponent.
02

Apply the Rule of Negative Exponents

The rule of negative exponents states that for any non-zero base, \(a^{-n} = \frac{1}{a^n}\). In this expression, applying the rule, the term \(-w^{-2}\) becomes \(-\frac{1}{w^2}\).
03

Simplify the Expression

The expression \(-\frac{1}{w^2}\) is the simplified form using only positive exponents. No further simplification is possible as all exponents are positive now.

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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 be a source of confusion, but they have a simple and elegant rule. A negative exponent indicates that the base of the power should be taken as the reciprocal. This means flipping the fraction. For example, if you have a number like \( w^{-2} \), it can be rewritten as \( \frac{1}{w^2} \). Essentially, each step towards understanding this concept involves reversing your perspective on numbers with negative exponents.

When dealing with negative exponents:
  • Identify the base and the negative exponent.
  • Transform the term with the negative exponent into its reciprocal.
  • Rewrite the expression with the positive exponent in the denominator.
By following these easy guidelines, you will have demystified negative exponents into manageable pieces.
Positive Exponents
While negative exponents express reciprocals, positive exponents come with a more straightforward interpretation. They simply indicate how many times you should multiply the base by itself. For instance, \( w^2 \) means \( w \times w \). Positive exponents are generally easier to handle in algebra since they don't involve the additional step of inversion.

When simplifying expressions using positive exponents:
  • The calculations are direct: just multiply the base by itself as many times as indicated by the exponent.
  • Positive exponents usually offer the simplest form of expressions; some expressions require converting negative exponents to positive for more streamlined results.
This straightforwardness is why expressions are often written in terms of positive exponents, providing clarity and ease of understanding.
Algebraic Simplification
Algebraic simplification is an essential skill when managing expressions with multiple operations. It involves minimizing the complexity of expressions while maintaining their original value. When dealing with exponents, simplification usually implies converting negative exponents into positive exponents.

To simplify algebraic expressions involving exponents:
  • Identify all parts of the expression and their exponents.
  • Convert negative exponents into positive by applying the reciprocal rule.
  • Look for opportunities to consolidate terms and reduce the expression to its simplest form.
In the given exercise, by converting \(-w^{-2}\) into \(-\frac{1}{w^2}\), we simplified the expression and ensured it only used positive exponents. This highlights the importance of understanding the rules and applying them correctly for clean and clear algebraic expressions.

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