Problem 62 - Algebra for College Students

Question

Simplify each expression. Express final results without using zero or negative integers as exponents. \(\frac{x^{-3} y^{-4}}{x^{2} y^{-1}}\)

Step-by-Step Solution

Answer

\( \frac{1}{x^5 y^3} \)

Step 1: Write the Given Expression

We are given the expression \( \frac{x^{-3} y^{-4}}{x^{2} y^{-1}} \). Our goal is to simplify it by eliminating zero or negative exponents.

Step 2: Eliminate Negative Exponents in the Numerator

In the numerator, we have \( x^{-3} y^{-4} \). Using the property \( a^{-n} = \frac{1}{a^n} \), we write this as \( \frac{1}{x^3 y^4} \).

Step 3: Rewrite the Whole Expression Using Fraction Properties

Substituting the rewritten numerator into the entire expression, we have: \( \frac{\frac{1}{x^3 y^4}}{x^2 y^{-1}} \). This is the same as multiplying by the reciprocal of the denominator: \( \frac{1}{x^3 y^4} \times \frac{1}{x^2 y^{-1}} \).

Step 4: Eliminate Negative Exponents in the Denominator

Rewrite \( y^{-1} \) in the denominator as \( \frac{1}{y} \). The expression becomes \( \frac{1}{x^3 y^4} \times \frac{y}{x^2} \).

Step 5: Simplify the Expression

Combine the fractions: \( \frac{y}{x^3 y^4 x^2} \). Simplify by multiplying the bases: \( = \frac{y}{x^{3+2} y^4} = \frac{y}{x^5 y^4} \).

Step 6: Further Simplify by Reducing Fractions

Reduce \( \frac{y}{y^4} \): Since \( y = y^1 \), the exponent is subtracted \( y^{1-4} = y^{-3} \). Thus, the expression is \( \frac{1}{x^5 y^3} \).

Step 7: Final Simplified Expression with Positive Exponents

We rewrite \( y^{-3} \) using positive exponents. So the expression simplifies to \( \frac{1}{x^5 y^3} \), which is already in suitable form with positive exponents.

Key Concepts

Negative Exponents

Negative exponents might seem a bit tricky, but they express an important mathematical concept. When you see a negative exponent like \( a^{-n} \), it's actually a way of writing the reciprocal of the base with a positive exponent. In simpler terms, \( a^{-n} = \frac{1}{a^n} \). This means you're flipping the position of the base in the fraction. Let's use the expression \( x^{-3} \) as an example. The negative exponent tells you to flip \( x^3 \) to the denominator, resulting in \( \frac{1}{x^3} \).
In algebra, changing negative exponents to their reciprocal form helps simplify expressions and solve problems more easily. Remember, always convert your expression to include only positive exponents for clarity and consistency. It's like cleaning up to make sure everything is in its right place!

Fraction Simplification

Simplifying fractions involves reducing the expression to its simplest form. It makes the expression easier to handle and understand.

For instance, when you're given a fraction like \( \frac{x^{-3}y^{-4}}{x^2y^{-1}} \), the key is to handle both the numerator and the denominator efficiently. Start by converting all negative exponents to positive ones by using the reciprocal rule.

• Numerator: convert \( x^{-3} \) to \( \frac{1}{x^3} \) and \( y^{-4} \) to \( \frac{1}{y^4} \).
• Denominator: change \( y^{-1} \) to \( \frac{1}{y} \).

Once both parts are rewritten with positive exponents, you multiply or divide to simplify further. Combining them together, you reach an expression where no more terms can be simplified, giving a clear and concise result.

Algebraic Expressions

In algebra, expressions can often seem like a puzzle until you break them down into understandable pieces. An algebraic expression uses numbers, letters (variables), and operations to represent a value or relationship. Taking the step-by-step approach simplifies even complex expressions.

Consider the expression \( \frac{1}{x^3 y^4} \times \frac{y}{x^2} \). During simplification, you combine like terms and adjust the fractions accordingly. You'll use laws of exponents for adding and subtracting exponents, always aiming to reorganize terms in a way that they have positive exponents.

• Combine like bases: add/subtract the exponents accordingly.
• For example: \( x^5 \) can emerge from \( x^{3+2} \).
• With the \( y \)'s, \( y\) over \( y^4 \) results in \( y^{-3} \), which simplification converts to \( \frac{1}{y^3} \).

By simplifying algebraic expressions this way, you ensure they are tidy and easy to interpret, making it simpler to spot patterns or to solve equations further along the way.