Problem 75 - Algebra for College Students

Question

Simplify each of the following. Express final results using positive exponents only. For example,\(\left(2 x^{\frac{1}{2}}\right)\left(3 x^{\frac{1}{3}}\right)=6 x^{\frac{5}{6}}\). \(\left(\frac{x^{2}}{y^{3}}\right)^{-\frac{1}{2}}\)

Step-by-Step Solution

Answer

The simplified expression is \( \frac{y^{\frac{3}{2}}}{x} \).

Step 1: Apply the Negative Exponent Rule

The given expression is \( \left( \frac{x^2}{y^3} \right)^{-\frac{1}{2}} \). Start by applying the negative exponent rule, which states that \( a^{-n} = \frac{1}{a^n} \). Thus, \( \left( \frac{x^2}{y^3} \right)^{-\frac{1}{2}} = \frac{1}{\left( \frac{x^2}{y^3} \right)^{\frac{1}{2}}} \).

Step 2: Simplify the Square Root of a Fraction

Now, express \( \left( \frac{x^2}{y^3} \right)^{\frac{1}{2}} \) as \( \frac{x^{2 \times \frac{1}{2}}}{y^{3 \times \frac{1}{2}}} \). This results in \( \frac{x^1}{y^{\frac{3}{2}}} \) or simply \( \frac{x}{y^{\frac{3}{2}}} \).

Step 3: Rewrite the Fraction with Positive Exponents

We now have \( \frac{1}{\left( \frac{x}{y^{\frac{3}{2}}} \right)} \). To get rid of the fraction in the denominator, multiply by its reciprocal: \( \frac{1}{\frac{x}{y^{\frac{3}{2}}}} = \frac{y^{\frac{3}{2}}}{x} \).

Step 4: Express Final Result

The simplified expression using positive exponents is \( \frac{y^{\frac{3}{2}}}{x} \).

Key Concepts

Understanding Negative Exponents

Negative exponents might seem tricky at first, but they're quite manageable once you get the hang of the concept. Essentially, a negative exponent represents the reciprocal of the base raised to the equivalent positive exponent. This means that if you have an expression like \( a^{-n} \), it is equivalent to \( \frac{1}{a^n} \).

For example, consider \( 2^{-3} \). To simplify, you follow the rule and rewrite it as \( \frac{1}{2^3} = \frac{1}{8} \). This tiny rule is a handy tool for turning negative exponents into something more familiar and easier to compute. Once you master it, handling expressions like \( \left( \frac{x^2}{y^3} \right)^{-\frac{1}{2}} \) becomes pretty straightforward. Just remember that negative exponents simply "flip" the fraction.

Using the Quotient Rule of Exponents

The quotient rule of exponents is a powerful tool for simplifying expressions involving division between similar bases. According to this rule, when you divide two powers with the same base, you simply subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \).

This rule is especially useful when you are working with fractions that have exponents. Take a look at the expression \( \frac{x^2}{y^3} \). If you have to raise this to any power, apply the power to both the numerator and the denominator separately, as shown in \( \left( \frac{x^2}{y^3} \right)^{\frac{1}{2}} = \frac{x^{2 \times \frac{1}{2}}}{y^{3 \times \frac{1}{2}}} \). This simplifies to \( \frac{x^1}{y^{\frac{3}{2}}} \).

Notice how the rule helps break down the expression into more manageable pieces. Simplifying fractions and mixed exponents becomes far less daunting when you consistently apply the quotient rule.

Simplification of Expressions

Simplifying expressions is all about making them more presentable or easier to work with by minimizing components like fractions and negative exponents. Once negative exponents are converted, and similar bases are handled via the quotient rule, the next step is often to express everything using positive exponents.

For instance, if you reach a point where you have an expression like \( \frac{1}{\left( \frac{x}{y^{\frac{3}{2}}} \right)} \), the goal is to rewrite it so there are no fractions in the denominator. You do this by multiplying by the reciprocal, leading to \( \frac{y^{\frac{3}{2}}}{x} \).

The process may seem complex initially, but by systematically applying rules and steps as shown with the expression \( \left( \frac{x^2}{y^3} \right)^{-\frac{1}{2}} \), simplification becomes achievable. Keep practicing these steps, and soon it will become second nature.