Problem 44 - Algebra A Combined Function

Question

Simplify each expression. Write each result using positive exponents only. $$ \frac{-5 x^{4} y^{5}}{15 x^{4} y^{2}} $$

Step-by-Step Solution

Answer

\(-\frac{1}{3} y^3\)

Step 1: Simplify the Coefficients

First, simplify the coefficient fraction \( \frac{-5}{15} \). Since both numbers have a common factor of 5, divide the numerator and the denominator by 5: \( \frac{-5}{15} = \frac{-1}{3} \). So, the expression becomes \( \frac{-1}{3} \times \frac{x^4 y^5}{x^4 y^2} \).

Step 2: Simplify the x Terms

Now, simplify the x terms by dividing \( x^4 \) by \( x^4 \). Remember that any number or variable divided by itself is 1, so this simplifies to 1: \( \frac{x^4}{x^4} = 1 \). This simplifies our expression to \( \frac{-1}{3} \times \frac{y^5}{y^2} \).

Step 3: Simplify the y Terms

For the y terms, use the law of exponents, which states \( \frac{y^a}{y^b} = y^{a-b} \). Here, \( a = 5 \) and \( b = 2 \), so this becomes \( y^{5-2} = y^3 \). Now the expression simplifies to \( \frac{-1}{3} y^3 \).

Key Concepts

Law of Exponents

Whenever you see exponents peeking out from polynomials, the Law of Exponents becomes a handy tool. This law essentially tells us how to handle expressions involving powers efficiently.
The basic rule to remember is: when dividing like bases, you simply subtract their exponents.

• If you have \(a^m\) divided by \(a^n\), the result is \(a^{m-n}\).
• This rule makes simplifying expressions swift and straightforward.

By grasping this concept, you can easily manipulate and reduce algebraic expressions.

Dividing Polynomials

Polynomials are like the building blocks of algebra, and dividing them is one of the fundamental operations you’ll need to master. When you're dividing polynomials like \(\frac{x^m y^n}{x^p y^q}\), each variable is handled independently based on their respective exponents.
Remember the key steps:

• Divide the coefficients first. Simplify them if there's a common factor.
• Use the Law of Exponents on each variable separately.

This ensures that your result is a neat, manageable expression, free from superfluous components.

Positive Exponents

Expressions should often be expressed with positive exponents to maintain consistency and clarity. Why use positive exponents? Well, they are far simpler to understand and work with.

• Having positive exponents means all factors are direct multiplications.
• The expression \(a^{-b}\) can be rewritten as \(\frac{1}{a^b}\), but this isn’t necessary if the result naturally yields positive exponents.

To keep things straightforward, avoid fractions inside exponents and ensure all powers remain positive after simplification.

Algebraic Fractions

Algebraic fractions vary slightly from regular fractions, as they involve variables. Solving them requires knowing how to handle both numbers and letters.

• Simplify the coefficients just like you would numerical fractions.
• Use exponent rules to simplify variables.

These steps transform algebraic fractions into simpler expressions, making calculations more manageable. With practice, these become as intuitive as handling regular fractions.