Problem 114 - College Algebra

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 is \(x^{-2} y^{8.25}\) (or \(x^{-2} y^{33/4}\) in fractional form).

Step 1: Handle the outer exponent

The first thing to do is handle the outer exponent, -4. By the power of a power rule in exponents, we know that \((a^n)^m = a^{n \cdot m}\). Thus, we get \((x^{\frac{1}{2}} y^{-\frac{7}{4}})^{-4} = x^{-2} y^{7}\).

Step 2: Handle the denominator

Next, we divide by \(y^{-\frac{5}{4}}\). Using the division rule for exponents, we obtain \(x^{-2} y^{7} \cdot y^{\frac{5}{4}}\, or \, x^{-2} y^{(\frac{7}{1}+\frac{5}{4})}\, which simplifies to \, x^{-2} y^{(\frac{33}{4})}\).

Step 3: Simplify

Finally, you can simplify the expression. You will end up with \(x^{-2} y^{8.25}\). Note: if instructed to keep the answer in fractional form, it would be appropriate to leave the exponent as \(y^{33/4}\).

Key Concepts

Power of a Power Rule

The power of a power rule is an important concept in algebra that simplifies the process of raising a power to another power. It states that when you have an exponent raised to another exponent, you multiply the exponents together. For example, \( (a^m)^n = a^{m \cdot n} \). This rule is particularly useful when dealing with complex expressions involving multiple layers of exponents. By applying this rule, we can easily transform expressions like \( (x^{\frac{1}{2}})^{-4} \) into \( x^{-2} \), as seen in the exercise you are working on. This not only makes the expressions more manageable but also sets the stage for further simplifications using other rules of exponents.

Division Rule for Exponents

The division rule for exponents states that when you divide two expressions with the same base, you subtract the exponent of the denominator from the exponent of the numerator, \( \frac{a^m}{a^n} = a^{m-n} \). This rule comes in handy when you need to simplify expressions involving division of exponential terms. Following this rule, if we take \( x^{-2} \cdot y^{7} \) and divide by \( y^{-\frac{5}{4}} \), it becomes \( x^{-2} y^{\frac{7}{1}+\frac{5}{4}} \) or \( x^{-2} y^{\frac{33}{4}} \), thereby simplifying the complex fraction into a single term with one base and one exponent.

Negative Exponents

Negative exponents indicate that the base is on the opposite side of a fraction. In other words, \( a^{-n} = \frac{1}{a^n} \). This implies that when you have a negative exponent, you can rewrite the expression as the reciprocal of the base raised to the positive exponent. It's an essential concept for simplifying expressions and solving equations. For instance, in the expression \( x^{-2} \) from the given problem, you can interpret this as \( \frac{1}{x^2} \) if you were to remove the negative exponent by transferring the base to the denominator side of a fraction.

Rational Exponents

Rational exponents represent roots and indicate exponentiation by a fraction. The expression \( a^{\frac{m}{n}} \) is equivalent to the nth root of \( a \) raised to the mth power or \( \sqrt[n]{a^m} \) or \( (\sqrt[n]{a})^m \). This means that you can rewrite expressions with fractional exponents as roots, providing another way to simplify complex numbers. A rational exponent can be both a tool for simplification and a means to represent more complicated expressions in a different form, which can be particularly useful in higher-level mathematics where roots are dealt with frequently.