Problem 68 - Algebra and Trigonometry

Question

\(47-72\) . Simplify the expression, and eliminate any negative exponent(s). $$ \left(\frac{x^{2} y}{2 y^{3}}\right)^{-2} $$

Step-by-Step Solution

Answer

\( \frac{4y^{4}}{x^{4}} \)

Step 1: Simplify the Fraction Inside the Parentheses

The expression given is \( \left(\frac{x^{2} y}{2 y^{3}}\right)^{-2} \). Start by simplifying the fraction inside the parentheses. Combine the terms with the same base:\[ \frac{x^{2} y}{2 y^{3}} = \frac{x^{2}}{2} \cdot \frac{y}{y^3} = \frac{x^{2}}{2} \cdot y^{-2}. \]

Step 2: Simplify Further Using Fraction and Negative Exponents Rules

Now, using the rules of exponents that state \( a^{-n} = \frac{1}{a^n} \) and \( \frac{a^m}{a^n} = a^{m-n} \), simplify further:\[ \frac{x^{2}}{2} \cdot y^{-2} = \frac{x^{2} y^{-2}}{2}. \]

Step 3: Apply the Negative Exponent

Apply the negative exponent to the fraction: \[ \left(\frac{x^{2} y^{-2}}{2}\right)^{-2} = \left(\frac{2}{x^{2} y^{-2}}\right)^{2}. \] This is based on \( \left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^{n} \).

Step 4: Expand the Fraction with the Power

Now, expand the fraction using the power outside:\[ \left(\frac{2}{x^{2} y^{-2}}\right)^{2} = \frac{2^{2}}{(x^{2})^{2} (y^{-2})^{2}}. \] This gives us:\[ \frac{4}{x^{4} y^{-4}}. \]

Step 5: Eliminate the Negative Exponent by Rewriting

Finally, eliminate the negative exponent in the expression by rewriting it:\[ \frac{4}{x^{4} y^{-4}} = \frac{4 imes y^{4}}{x^{4}} = \frac{4y^{4}}{x^{4}}. \]

Key Concepts

Simplifying Expressions

Simplifying expressions is a crucial math skill that makes equations more manageable. It involves transforming complex expressions into simpler ones without changing their value. The key to simplifying is to identify and apply mathematical properties effectively. In the case of the expression \(\left(\frac{x^{2} y}{2 y^{3}}\right)^{-2}\), we start by simplifying the fraction within the parentheses. This involves combining like terms and removing unnecessary complexity.To do this, we look at similar bases in numerators and denominators and apply the division rule of exponents, which states \( \frac{a^m}{a^n} = a^{m-n} \). In our expression, \(\frac{x^{2} y}{2 y^{3}}\) becomes \(\frac{x^{2}}{2} \cdot y^{-2}\) after simplifying. This reduction helps make subsequent calculations, like applying exponents, much easier.

Negative Exponents

Negative exponents can initially seem confusing, but they're straightforward once you understand the rule: \(a^{-n} = \frac{1}{a^n}\). This means a negative exponent signifies reciprocal action. For example, in the expression \(\left(\frac{x^{2} y^{-2}}{2}\right)^{-2}\), applying the negative exponent flips the fraction.When dealing with negative exponents within a term, like in our example \( y^{-2} \), the rule can be used to rewrite these terms as reciprocals, which shifts them to the denominator or numerator appropriately. Therefore, \( y^{-2} \) becomes \( \frac{1}{y^2} \) when rewritten in the denominator, helping transition from complex fractions to manageable forms that can be expanded or further simplified.

Fractional Exponents

Fractional exponents involve raising numbers to a fractional power, which is equivalent to taking roots. For example, \( a^{\frac{m}{n}} \) is the same as \( \sqrt[n]{a^m} \). They provide a more flexible method of expressing powers and roots in equations. Our example deals more with regularly utilizing powers, but understanding fractional exponents builds the foundation for managing expressions later.When expanding \( \left(\frac{2}{x^{2} y^{-2}}\right)^{2} \), we're effectively distributing the power across the fraction, resulting in each component being squared. This means \( \frac{2}{x^{2}y^{-2}} \) becomes \( \frac{2^2}{(x^{2})^2(y^{-2})^2} \). Fractional exponents can similarly distribute across terms, just as integers.