Problem 36 - Beginning and Intermediate Algebra

Question

Simplify. $$\frac{x^{3} y^{-3}}{x^{-1} y^{2}}$$

Step-by-Step Solution

Answer

The simplified expression is \(x^{4} \cdot y^{-5}\).

Step 1: Identify the given expression

The given expression is: \(\frac{x^{3} y^{-3}}{x^{-1} y^{2}}\)

Step 2: Use exponent rules for division

To simplify the expression, use the exponent rule for division, which states that when dividing the same base with different exponents, subtract the exponents: \(\frac{x^{3} y^{-3}}{x^{-1} y^{2}} = x^{3 - (-1)} \cdot y^{-3 - 2}\)

Step 3: Simplify by adding exponents

Perform the subtraction in the exponents: \(x^{3 - (-1)} \cdot y^{-3 - 2} = x^{3 + 1} \cdot y^{-3 - 2} = x^{4} \cdot y^{-5}\)

Step 4: Final simplified expression

The simplified expression is: \(x^{4} \cdot y^{-5}\)

Key Concepts

Simplifying Exponents

When faced with a complex expression involving exponents, the goal is often to simplify it. Simplifying exponents makes the expression easier to work with and understand. Here's how it's done:

• Identify similar bases. Begin by looking for terms in your expression that have the same base. Exponents with the same base can be simplified using specific rules.
• Apply the necessary exponent rules. To simplify, use rules such as product and quotient rules for exponents.
• Streamline your expression. Simplification aims to reduce the number of terms in the expression while keeping the mathematical meaning the same.

By properly applying these steps, you'll find it easier to manipulate algebraic expressions and solve equations.

Exponents in Algebra

Exponents play a crucial role in algebra. They indicate how many times a number, known as the base, is multiplied by itself. Understanding exponents is essential for working with more advanced mathematical concepts. In algebra:

• Exponents are shorthand for repeated multiplication. For example, the expression \( x^3 \) means \( x \times x \times x \).
• Exponent rules simplify expressions. The main exponent rules include multiplication (adding exponents), division (subtracting exponents), and power of a power (multiplying exponents).
• Algebraic expressions frequently use exponents. Polynomials, for example, often include terms with exponents.

By mastering exponent rules, you'll be able to tackle algebraic problems more efficiently, simplifying the process of solving equations and inequalities.

Negative Exponents

Negative exponents may at first seem confusing, but they follow clear rules. A negative exponent means that the base is on the opposite side of the fraction line. Here's how it works:

• Turn negative exponents into fractions. For example, \( x^{-3} \) becomes \( \frac{1}{x^3} \).
• Treat negative exponents as reciprocals. Inverting the base nullifies the negative sign on the exponent.
• Simplify mixed expressions. When an expression includes both positive and negative exponents, simplify each part based on its exponent.

Understanding negative exponents is essential for simplifying expressions with mixed signs and performing accurate calculations. By recognizing their role, you'll enhance your algebra skills substantially.