Problem 23 - College Algebra

Question

Simplify each exponential expression $$ x^{-2} y $$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(y/x^2\).

Step 1: Rewrite the negative exponent

To simplify the expression, first rewrite the term with a negative exponent. Remember that the negative exponent indicates a reciprocal. So, substitute \(x^{-2}\) with \(1/x^2\). Our expression now reads \(1/x^2*y\) (or alternatively written as \(y/x^2\)).

Step 2: Final form of the expression

There is no further simplification possible in this expression. Therefore, the expression in its simplest form is \(y/x^2\).

Key Concepts

Negative Exponents Explained

Negative exponents can sometimes be confusing at first, but they simply tell us to take the reciprocal of the base raised to the positive of that exponent. For example, if you encounter a term like \( x^{-2} \), it might initially seem complex. However, you only need to convert it into its reciprocal. So, \( x^{-2} \) becomes \( 1/x^2 \). The concept of negative exponents is straightforward:

• They convert multiplication into division.
• The base stays the same; only the position (i.e., numerator or denominator) changes.

By understanding this, you're equipped to handle many exponential expressions more effortlessly.

Simplification of Expressions: Making it Simple

Simplification is a process of making mathematical expressions easier to work with. Here, our goal is to reduce complexity so that mathematical computations and comparisons become clearer. Simplifying the expression \( x^{-2}y \) requires handling each component correctly:

• The term \( x^{-2} \) becomes \( 1/x^2 \) when simplified using the negative exponent rule.
• Combining this with \( y \), there is no common factor to further simplify. You write it as \( y/x^2 \).

This simplified form is easier to understand and work with. By simplifying, you prevent errors in further calculations and ensure clarity in solving larger equations.

Reciprocal of Exponents: Turning Topsy-Turvy

When we deal with negative exponents, recognizing the reciprocal relationship is key. But what does reciprocation mean exactly? Think of reciprocals as flipping numbers across the fraction line. Applying this to exponents means:

• \( x^{-a} \) becomes \( 1/x^a \), effectively moving the term from the numerator to the denominator.
• Conversely, \( 1/x^{-a} \) shifts \( x^a \) back to the numerator, transforming the expression profoundly.

This reciprocal nature allows us to convert otherwise complex expressions into forms that are easier or sometimes even essential, particularly in solving equations. Keep this notion in your toolbox as it’s a frequent necessity in handling mathematics involving powers and roots.