Problem 70 - Algebra 2

Question

Simplify each expression. Assume that all variables are positive. $$\frac{x^{\frac{1}{2}} y^{-\frac{1}{3}}}{x^{\frac{3}{4}} y^{\frac{1}{2}}}$$

Step-by-Step Solution

Answer

After the simplification, the final simplified form of the given expression is \( \frac{y^{\frac{5}{6}}}{x^{\frac{1}{4}}} \)

Step 1: Use the rule of exponents to simplify the denominator and the numerator separately

The given expression can be simplified by applying the laws of exponents to both the numerator and the denominator separately: \(x^{\frac{1}{2}} y^{-\frac{1}{3}}\) becomes \(x^{\frac{1}{2}} \times \frac{1}{y^{\frac{1}{3}}}\) and \(x^{\frac{3}{4}} y^{\frac{1}{2}}\) remains the same.

Step 2: Combine like terms

Now, unite the like terms, i.e., the terms involving the same variable, and subtract the exponent of the divisor from the exponent of the dividend. Hence, \( \frac{x^{\frac{1}{2}}}{x^{\frac{3}{4}}} \) becomes \( x^{\frac{1}{2}-\frac{3}{4}}\) and \( \frac{1}{y^{\frac{1}{3}}} \) divided by \( y^{\frac{1}{2}} \) becomes \( y^{\frac{1}{3}-(-\frac{1}{2})} \)

Step 3: Simplify the expressions

Now, perform the operations inside the parentheses to simplify the expressions. The simplification will give us \( x^{-\frac{1}{4}} \) and \( y^{\frac{5}{6}} \)

Key Concepts

Laws of Exponents

The Laws of Exponents are fundamental rules that help us deal with expressions involving powers of the same base. These laws make manipulation of exponents straightforward and consistent. Let's dive into some of the most commonly used laws:

• Product of Powers Rule: When multiplying two exponents with the same base, you keep the base and add the exponents. For instance, \( a^m imes a^n = a^{m+n} \).
• Quotient of Powers Rule: When dividing two exponents with the same base, you keep the base and subtract the exponents. For example, \( \frac{a^m}{a^n} = a^{m-n} \).
• Power of a Power Rule: When raising an exponent to another power, you multiply the exponents. For example, \( (a^m)^n = a^{m \times n} \).

These rules are essential when simplifying complex expressions like the one in our problem. By applying these laws correctly, simplifying expressions becomes an orderly task, making calculations easier and faster.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form while keeping the expression's value unchanged. The process generally follows a step-by-step approach:

• Identify Like Terms: Look for terms in the expression that have the same base.
• Apply Exponent Laws: Use the laws of exponents to combine like terms. This may involve adding, subtracting, or multiplying the exponents.
• Simplify Further: Whenever possible, simplify any numerical calculations or reduce fractions.

In the exercise example, the steps include applying the quotient of powers rule and correctly handling negative exponents while assuming all variables are positive. Each step carefully reduces the expression's complexity without changing its fundamental value. Thus, we arrive at simpler expressions, such as \( x^{-rac{1}{4}} \) and \( y^{\frac{5}{6}} \), efficiently.

Rational Exponents

Rational exponents are a way of expressing roots as powers. An exponent expressed as a fraction means you are dealing with both a root and a power. For instance, \( a^{\frac{m}{n}} \) translates to the \( n \)-th root of \( a^m \) or \( (\sqrt[n]{a})^m \). This notation is particularly handy when you want to express roots using the same exponent rules applied to whole numbers:

• Conversion Between Roots and Fractional Powers: For instance, \( a^{\frac{1}{2}} \) is the same as \( \sqrt{a} \), and \( a^{\frac{1}{3}} \) coincides with \( \sqrt[3]{a} \).
• Using Exponent Laws with Rational Exponents: Applying the laws of exponents, you can simplify complex expressions with rational exponents in a similar fashion to whole numbers.

In the given exercise, rational exponents like \( x^{\frac{1}{2}} \) and \( y^{-\frac{1}{3}} \) appear. Understanding how to manipulate these using exponent laws is crucial for solving and simplifying expressions accurately, achieving results such as \( x^{-rac{1}{4}} \) and \( y^{\frac{5}{6}} \).