Problem 23 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The simplified exponential expression is \(\frac{y}{x^2}\)

Step 1: Recall Negative Exponent Rule

Remember that a term with a negative exponent should be written on the opposite side of the fraction line with a positive exponent.

Step 2: Apply Negative Exponent Rule

Apply the negative exponent rule. The x^(-2) term moves to the denominator of the fraction so it becomes: \(\frac{y}{x^2}\).

Step 3: Write the Simplified Expression

No more simplification can be done, so leave the expression in its current form, which is \(\frac{y}{x^2}\)

Key Concepts

Exponential Expressions

Exponential expressions are a powerful way to represent repeated multiplication. When we see something like \( x^3 \), it means \( x \) multiplied by itself three times, so \( x \times x \times x \). Exponents can be whole numbers, fractions, or even negative numbers. When dealing with any kind of exponential expression, it is essential to understand the rules governing exponents, as this will allow us to simplify them effectively. In our problem, we are dealing with the exponential expression \( x^{-2} y \). It's important to note the negative exponent in \( x^{-2} \), which indicates a specific operation for simplification.

Fraction Simplification

Fraction simplification involves reducing a fraction to its simplest form. In the context of exponential expressions, simplification often involves moving terms around using the rules of exponents. When expression terms are in fraction form, each component within the numerator or denominator is evaluated to see if any can be simplified further.
In our exercise, once we apply the negative exponent rule to \( x^{-2} y \), we move \( x^{-2} \) from the numerator to the denominator, thereby turning it into \( x^2 \) in the denominator due to the positive conversion. The simplified exponential expression becomes \( \frac{y}{x^2} \). In this case, no further simplification can occur because the fraction is already in its simplest form by placing \( x^2 \) in the denominator.

Negative Exponent Rule

The negative exponent rule is a key principle that helps us understand how to handle negative exponents in expressions. It states that any base raised to a negative exponent can be turned into a positive exponent by switching its position from the numerator to the denominator, or vice versa. Essentially, \( a^{-n} = \frac{1}{a^n} \).
This rule is beneficial because it allows us to convert complex expressions into more manageable forms. Applying the negative exponent rule in our original problem \( x^{-2} y \), we see that \( x^{-2} \) becomes \( \frac{1}{x^2} \) when moved to the denominator, making the expression \( \frac{y}{x^2} \).
Understanding this rule is crucial for simplifying and solving exponential expressions correctly, ensuring that no mathematical mistakes occur when moving terms around.