Problem 32 - Elementary Algebra

Question

Divide. (Assume all denominators are nonzero.) $$ 5 y 410(3 y-5) 2 \div 10 y 52(3 y-5) 3 $$

Step-by-Step Solution

Answer

The expression simplifies to \( \frac{1}{2y(3y-5)} \).

Step 1: Write the Division as a Fraction

The division of expressions can be written as a fraction. For the given problem, express it as: \[ \frac{5y^4(3y-5)^2}{10y^5(3y-5)^3} \].

Step 2: Simplify the Coefficients

Look at the coefficients (numerical values) in the numerator and the denominator. You have 5 in the numerator and 10 in the denominator. Simplify \( \frac{5}{10} \) to \( \frac{1}{2} \).

Step 3: Simplify the 'y' Terms

To simplify the \( y \) terms, subtract the powers of \( y \): \( y^4 \) in the numerator and \( y^5 \) in the denominator. \( y^{4-5} = y^{-1} \), which can be written as \( \frac{1}{y} \).

Step 4: Simplify the Polynomial Factors (3y-5)

Observe the polynomial factors (3y-5). There are two copies in the numerator and three in the denominator: \( (3y-5)^2 \) and \( (3y-5)^3 \) respectively. Simplify \( \frac{(3y-5)^2}{(3y-5)^3} \) to \( \frac{1}{3y-5} \) by subtracting the exponents following the rules of exponents.

Step 5: Combine All Simplified Results

Combine all simplified parts: \( \frac{1}{2} \), \( \frac{1}{y} \), \( \frac{1}{3y-5} \). Therefore, the expression simplifies to \( \frac{1}{2y(3y-5)} \).

Key Concepts

Fraction Simplification

Fraction simplification is all about making a fraction as simple as possible. Start by writing division problems as fractions. This is because fractions are essentially division problems written in a special way. For instance, when you see something like \( \frac{5y^4(3y-5)^2}{10y^5(3y-5)^3} \), it means you're dividing everything in the numerator by everything in the denominator.

To simplify a fraction, follow these easy steps:

• Look at each part of the fraction separately: coefficients (numbers), variables like \(y\), and any other expressions.
• For coefficients, divide them as you would with normal numbers. Here, divide 5 by 10 to get 0.5 or \(\frac{1}{2}\).
• If variables or expressions are repeated in both the numerator and the denominator, simplify them by dividing or canceling them out, as shown in the exercise solution.

By breaking it down, you make fractions more manageable and less intimidating!

Polynomial Factor Simplification

Polynomial factor simplification involves making complex polynomial expressions more straightforward. Imagine polynomials as collections of terms, like a combined grocery list that you need to sort and simplify.

Let's tackle polynomial expressions that have common factors, like \((3y - 5)^2\) over \((3y - 5)^3\). Both expressions have the same base, \((3y - 5)\), but with different exponents (2 and 3 in this case).

To simplify polynomial factors:

• Identify common polynomial (repeating expression) parts in both the numerator and denominator.
• Use the rule of subtracting exponents to simplify. Subtract the smaller exponent from the larger one: \(3 - 2 = 1\), leaving you with \(\frac{1}{3y-5}\).

Simplifying polynomials this way allows you to condense lengthy expressions into neat, manageable forms.

Rules of Exponents

Understanding the rules of exponents is crucial in simplifying algebraic expressions. Exponents tell you how many times a number or variable is used as a factor in a multiplication.

Here are a few key rules you should always keep in mind:

• Multiplying Like Bases: When multiplying two expressions with the same base, add their exponents. For example, \(y^a \times y^b = y^{a+b}\).
• Dividing Like Bases: When dividing, subtract the exponents: \(y^a / y^b = y^{a-b}\).
• Zero Exponent Rule: Any number raised to the power of zero is 1: \(y^0 = 1\).

In the original exercise, simplifying \(y^4\) over \(y^5\) by subtracting the exponents gives you the result \(y^{-1}\), which means \(\frac{1}{y}\).

Mastering these rules can significantly ease the journey through algebraic problems, making complex equations far simpler to solve.