Problem 114 - Algebra and Trigonometry

Question

Simplify each expression. Assume that all variables represent positive numbers. $$ \left(\frac{x^{\frac{1}{2}} y^{-\frac{7}{4}}}{y^{-\frac{5}{4}}}\right)^{-4} $$

Step-by-Step Solution

Answer

The simplified expression of \(\left(\frac{x^{\frac{1}{2}} y^{-\frac{7}{4}}}{y^{-\frac{5}{4}}}\right)^{-4}\) is \(1/(x^2 y^{12})\).

Step 1: Break down the inner part of the parentheses

First, divide \(x^{\frac{1}{2}}\) by \(y^{-\frac{5}{4}}\) and multiply this division with \(y^{-\frac{7}{4}}\). Use the exponent rule \(a^b / a^c = a^{b-c}\) and \(a^{-b}=1/a^b\). As a result, the expression obtains the form of \((x^{\frac{1}{2}} y^{\frac{7}{4} - -\frac{5}{4}})^{-4}\), which simplifies to \((x^{\frac{1}{2}} y^3)^{-4}\).

Step 2: Distribute the exponent term outside the parenthesis to each term within

Spread out the outer exponent of \(-4\) to each term within the parentheses. Use the rule \((ab)^c = a^c * b^c\). The expression becomes \((x^{\frac{1}{2}})^{-4} * (y^3)^{-4}\), which can be simplified to \(x^{-2} * y^{-12}\).

Step 3: Convert the negative exponents into positive ones

The final step is to convert the negative exponents into positive ones. This is done based on the rule \(a^{-b} = 1/a^b\). Therefore, \(x^{-2}*y^{-12}\) becomes \(\frac{1}{x^2*y^{12}}\) or \(1/(x^2 y^{12})\).