Problem 49 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The simplified expression is \(- \frac{1}{w^2}\).

Step 1: Identify the Expression

The given expression is \(-w^{-2}\). Our objective is to rewrite this expression using positive exponents.

Step 2: Rewrite Using Positive Exponents

To rewrite \(-w^{-2}\) using positive exponents, recall that \(a^{-m} = \frac{1}{a^m}\). This means that \(w^{-2} = \frac{1}{w^2}\), and the negative sign indicates multiplication by -1.

Step 3: Simplify the Expression

Applying the rule from the previous step, rewrite the expression: \(-w^{-2} = - \frac{1}{w^2}\). This is the expression simplified with a positive exponent.

Key Concepts

Positive Exponents

Positive exponents are a way to represent repeated multiplication of a base number. When you see an exponent, it tells you how many times you multiply the base number by itself. For example, in the expression \(w^2\), the base \(w\) is multiplied by itself 2 times: \(w \times w\). Positive exponents are straightforward, and writing them down is simple. This form is crucial because it keeps things neat and easy to understand, especially when simplifying expressions.

• Positive exponents indicate repeated multiplication.
• They make expressions cleaner and more manageable.
• For \(a^m\), the base \(a\) is multiplied by itself \(m\) times.

Negative Exponents

Negative exponents might initially seem confusing, but they have a simple and elegant logic. A negative exponent tells us that instead of multiplying, we are going to divide. Specifically, \(a^{-m}\) means \(\frac{1}{a^m}\).
This idea transforms the calculation into a fraction, effectively turning the exponent positive while placing the base in the denominator.

• Negative exponents represent division by the base.
• The expression \(w^{-2}\) converts to \(\frac{1}{w^2}\).
• This technique allows all exponents to be positive, simplifying further operations.

Algebraic Manipulation

Algebraic manipulation involves a series of techniques to transform expressions into more manageable forms. When working with exponents, understanding how to shift between positive and negative exponents is key. For example, given the task to simplify \(-w^{-2}\), we start by rewriting the negative exponent as a fraction, i.e., \(w^{-2} = \frac{1}{w^2}\).
Then, maintain the negative sign outside to reflect multiplication by \(-1\), resulting in \(-\frac{1}{w^2}\).
Such manipulations help us see complicated expressions in a new, simplified light.

• Use known exponent rules to transform expressions.
• Rewriting helps reveal a clearer path to simplification.
• Always maintain accurate signs for clarity and correctness.