Problem 117 Raising a Product or a Quotient ... [FREE SOLUTION] (2024)

Short Answer

Negative Exponents

Negative exponents can be initially confusing, but they follow simple rules. A negative exponent means you take the reciprocal of the base and then apply the positive exponent. For instance, if you have a negative exponent like \(a^{-n}\), it translates to \(\frac{1}{a^n}\). So, \(5^{-2}\) becomes \(\frac{1}{5^2} = \frac{1}{25}\). This rule is crucial in the given problem where you have multiple terms with negative exponents that need to be simplified inside the parenthesis. Move the terms with negative exponents to the denominator if they are in the numerator and to the numerator if they are in the denominator. This step dramatically simplifies the expressions.

Simplifying Fractions

Simplifying fractions involves reducing the fraction to its simplest form without changing its value. In the context of exponents, simplifying means handling each component thoroughly before making it part of a larger expression. For example, in \(\frac{5 x^2}{2 y^{11}}\), we ensure \(5 x^2\) and \(2 y^{11}\) have no simplifications left. Breaking down complex expressions and individually simplifying them makes managing them more straightforward. In the solution, terms with zero exponents convert to 1, and negative exponents are moved across the fraction to become positive.

Power Rules for Exponents

Power rules help in handling expressions raised to exponents. The basic rule \( (a^m)^n = a^{m \times n} \) tells us how to manage exponents inside and outside parentheses. For example, in \( (x^2)^{-2} \), you multiply exponents: \( x^{2 \times -2} = x^{-4} \). Similarly, when applying exponents to a product or quotient, we multiply each term inside by the exponent. This makes complex expressions like \( \left( \frac{5x^2}{2y^{11}} \right)^{-2} \) manageable. Applying the power rule individually to the numerator and the denominator simplifies each part before recombining them.

Algebraic Manipulation

Algebraic manipulation refers to reworking terms and expressions to simplify or solve them. This step often involves combining terms, factoring, expanding, and rearranging. In this problem, after applying the negative exponent outside the parenthesis, the challenge is to simplify further. We decompose complicated terms by using known algebraic rules. We turn \( \frac{\frac{1}{25} x^{-4}}{ \frac{1}{4} y^{-22}}= \frac{4}{25} \times y^{22} \times x^{-4} \). Evaluating each part’s exponents and coefficients provides the final form. The accuracy in each step determines the correct final simplified result.