Problem 89 - College Algebra with Modeling and Visualization

Question

Simplify the expression and write it with rational exponents. Assume that all variables are positive. $$ \left(\frac{x^{2}}{y^{6}}\right)^{-1 / 2} $$

Step-by-Step Solution

Answer

The expression simplifies to \( \frac{y^3}{x} \).

Step 1: Apply the Power of a Quotient Rule

The expression given is \( \left(\frac{x^{2}}{y^{6}}\right)^{-1/2} \). According to the power of a quotient rule, \( \left(\frac{a}{b}\right)^m = \frac{a^m}{b^m} \). Applying this rule, we have \( \frac{(x^2)^{-1/2}}{(y^6)^{-1/2}} \).

Step 2: Apply the Negative Exponent Rule

The next step is to apply the negative exponent rule which states that \( a^{-m} = \frac{1}{a^m} \). Applying this rule, we get \( \frac{1}{x^{2 \times (-1/2)}} \) and \( \frac{1}{y^{6 \times (-1/2)}} \).

Step 3: Simplify the Exponents

Calculate the exponents in both the numerator and denominator. For the numerator: \( x^{2 \times (-1/2)} = x^{-1} \). For the denominator: \( y^{6 \times (-1/2)} = y^{-3} \). Therefore, we have \( \frac{x^{-1}}{y^{-3}} \).

Step 4: Simplify the Expression with Positive Exponents

Rewrite the expression using positive exponents. This involves flipping the expressions with negative exponents: \( x^{-1} = \frac{1}{x} \) and \( y^{-3} = \frac{1}{y^3} \). Thus, \( \frac{x^{-1}}{y^{-3}} = y^3 x^{-1} \). To keep the exponents rational, this simplifies to \( \frac{y^3}{x} \).

Key Concepts

Power of a Quotient Rule

When working with expressions in the form of fractions, the power of a quotient rule becomes extremely useful. The rule can be stated as \( \left( \frac{a}{b} \right)^m = \frac{a^m}{b^m} \). This means if you have a fraction raised to a power, you apply that power to both the numerator and the denominator separately. This rule is particularly helpful for expressions involving rational exponents, as rational exponents are just fractions themselves that indicate roots and powers.

• If you see a fraction like \( \left( \frac{x^2}{y^6} \right)^{-1/2} \), you apply \(-1/2\) to both \(x^2\) and \(y^6\) individually.
• This converts the expression into \( \frac{(x^2)^{-1/2}}{(y^6)^{-1/2}} \).
• By breaking it down into separate parts, the overall problem becomes more manageable.

It's like giving each part of the fraction its special little power to work with. Remember, this rule only applies when the entire fraction is raised by an exponent.

Negative Exponent Rule

Negative exponents can seem intimidating at first, but they are not as scary as they may appear. The negative exponent rule is stated as \( a^{-m} = \frac{1}{a^m} \). Simply put, a negative exponent moves the base to the opposite part of the fraction, making it positive.

• Think of \( a^{-m} \) as \( \frac{1}{a^m} \). The negative sign tells you to "flip" the base to the denominator.
• For instance, if you have \( (x^2)^{-1/2} \), you can rewrite it as \( \frac{1}{x^{2 \times (-1/2)}} \).
• The same goes for \( (y^6)^{-1/2} \), turning it into \( \frac{1}{y^{6 \times (-1/2)}} \).

By applying this rule, expressions become much simpler to handle, leading to easier calculation and simplification of complex expressions.

Simplifying Expressions

The final step in working with exponents is often to simplify the expression, making it easier to understand or use in further calculations. This especially involves working with the results of applying the power of a quotient and the negative exponent rules. Simplifying helps in converting expressions with negative exponents to positive ones whenever possible.

• To simplify \( \frac{x^{-1}}{y^{-3}} \), consider flipping both exponents by using the rule \( a^{-m} = \frac{1}{a^m} \).
• Therefore, \( x^{-1} \) becomes \( \frac{1}{x} \), and \( y^{-3} \) becomes \( \frac{1}{y^3} \).
• Thus, \( \frac{x^{-1}}{y^{-3}} = y^3 \cdot \frac{1}{x} \), which simplifies to \( \frac{y^3}{x} \).

The goal is to arrange the expression in the simplest and most concise form, often converting negative exponents into positive ones for clearer interpretations. This ending step is crucial as it checks if all the steps were correctly applied and leaves the expression in a friendly format for additional math operations.