Problem 33 - Algebra 2

Question

Simplify. Assume that no variable equals 0. $$ \left(\frac{x}{y^{-1}}\right)^{-2} $$

Step-by-Step Solution

Answer

The expression simplifies to \( \frac{1}{x^{2}y^{2}} \).

Step 1: Apply the Negative Exponent Rule

Start by rewriting the expression \( y^{-1} \) using the negative exponent rule: \( y^{-1} = \frac{1}{y} \). So, \( \frac{x}{y^{-1}} \) becomes \( \frac{x}{\frac{1}{y}} \).

Step 2: Simplify the Complex Fraction

To simplify \( \frac{x}{\frac{1}{y}} \), you multiply by the reciprocal of the denominator. Therefore, \( \frac{x}{\frac{1}{y}} \) simplifies to \( x \times y = xy \).

Step 3: Apply the Exponent to the Simplified Expression

Now that we have \( (xy) \), apply the exponent \(-2\) from the original expression to get \((xy)^{-2}\).

Step 4: Apply the Negative Exponent on Product Rule

Use the negative exponent rule: \( (ab)^{-n} = \frac{1}{(ab)^{n}} \). For \((xy)^{-2}\), we have \( \frac{1}{(xy)^{2}} \).

Step 5: Expand the Squared Product

Finally, apply the exponent to each factor in the numerator: \((xy)^{2} = x^{2}y^{2}\). Therefore, \( \frac{1}{(xy)^{2}} = \frac{1}{x^{2}y^{2}} \).

Key Concepts

Negative Exponents

Negative exponents can seem intimidating at first, but they are just another way to express division. In math, when you see a negative exponent, like in the expression \( y^{-1} \), it means you need to take the reciprocal or "flip" the base. So, for \( y^{-1} \), it is equal to \( \frac{1}{y} \). This transformation is crucial when simplifying expressions and understanding how to handle variables in an algebraic context.

• A negative exponent indicates a reciprocal.
• The operation changes the position of the base between numerator and denominator.

Understanding the nature of negative exponents allows you to manipulate expressions easily, such as in \( \frac{x}{y^{-1}} \), which can be rewritten to \( \frac{x}{\frac{1}{y}} \). As you get comfortable with this flipping concept, it becomes much easier to simplify complex algebraic expressions without getting stuck.

Fraction Simplification

Fraction simplification is a fundamental skill in algebra and mathematics. It involves rewriting fractions in a simpler, more understandable form. When faced with a complex fraction, such as \( \frac{x}{\frac{1}{y}} \), the key is to multiply by the reciprocal of the denominator. This step effectively eliminates the fraction within a fraction.

• Identify the reciprocal of the denominator.
• Multiply the numerator by this reciprocal.

In our example, multiplying \( x \) by the reciprocal of \( \frac{1}{y} \) gives us \( xy \). This approach simplifies the expression significantly by removing the inner fraction and providing a clearer path to apply subsequent operations. This technique of simplifying fractions is not just useful but crucial when tackling more complex algebraic problems.

Exponent Rules

Understanding the rules of exponents is essential when dealing with expressions involving powers. One of the key exponent rules is how to handle expressions like \( (ab)^{-n} \), which transforms into \( \frac{1}{(ab)^n} \). This rule allows us to handle negative exponents across multiple factors, simplifying expressions that would otherwise be challenging.

• Apply the exponent to each factor in a product.
• Use the negative exponent rule by flipping the expression.

When using exponent rules, always ensure you apply the power to both parts of a product independently, as shown in the last step where \( (xy)^{-2} \) becomes \( \frac{1}{x^2y^2} \). Ensuring each factor is treated correctly maintains the integrity and equality of the expression as you simplify it. Mastering these rules makes tackling math problems quicker and much less daunting.