Problem 14 - Algebra 2

Question

Divide. State any restrictions on the variables. $$ \frac{3 x^{3}}{5 y^{2}} \div \frac{6 x^{5}}{5 y^{3}} $$

Step-by-Step Solution

Answer

The result of the division is \( \frac{y}{2x^{2}} \). The denominator cannot be equal to zero, so the only restriction on the variable is that \(x\) cannot equal 0.

Step 1: Rewrite the Division as Multiplication

The division operation between two fractions can be reinterpreted as multiplication by the reciprocal of the second fraction. \n \( \frac{3 x^{3}}{5 y^{2}} \div \frac{6 x^{5}}{5 y^{3}} \) can be rewritten as \( \frac{3 x^{3}}{5 y^{2}} * \frac{5 y^{3}}{6 x^{5}} \)

Step 2: Simplify the Resulting Multiplication

Now, multiply the two fractions and reduce the result to its simplest form. The multiplication of the two fractions is: \n \( \frac{3 * 5 * x^{3} * y^{3}}{5 * 6 * y^{2} * x^{5}} \) \n By cancelling out the common terms in the numerator and the denominator, we get: \n \( \frac{y}{2x^{2}} \)

Step 3: Find the Variable Restrictions

Setting the denominator equal to zero and solving for the variable will find the restrictions on the variable. Here denominator is \(2x^{2}\). If we set this equal to zero, we get \(2x^{2} = 0\), which simplifies to \(x = 0\) only.

Key Concepts

Fraction Multiplication

When working with division between two fractions, one of the most important concepts is recognizing that division can be transformed into multiplication. To achieve this, multiply by the reciprocal.
This means flipping the numerator and the denominator of the second fraction before proceeding.

• The original problem stated: \[ \frac{3 x^{3}}{5 y^{2}} \div \frac{6 x^{5}}{5 y^{3}} \]
• By taking the reciprocal, it becomes: \[ \frac{3 x^{3}}{5 y^{2}} \times \frac{5 y^{3}}{6 x^{5}} \]

This approach not only simplifies the process but is also the cornerstone of dealing with operations involving fractions within algebra.
With this newfound expression, you can more easily tackle the multiplication process and apply any further simplifications.

Variable Restrictions

Understanding and identifying variable restrictions is crucial in algebra. Variable restrictions delimit the values for which an expression is defined.
In polynomial division, you must always ensure that the denominator does not equate to zero, as this would make the expression undefined.
Look at the simplified fraction we arrived at: \[ \frac{y}{2x^{2}} \]

• The denominator here is \(2x^{2}\), and to find restrictions, you set \(2x^{2} = 0\).
• Simplifying gives \(x = 0\), indicating \(x\) cannot be zero in the expression.

These restrictions help ensure that your calculations and results remain valid. Always keep an eye out for similar scenarios, as overlooking them can lead to incorrect solutions or interpretations.

Simplifying Algebraic Expressions

After transforming division into multiplication, the next vital step in solving the problem is simplifying the resulting expression.
Simplification involves reducing an expression to its most concise form without changing its value or meaning.
Using the resultant multiplication from the exercise:

• First, recognize like terms in the numerator and denominator: \[ \frac{3 * 5 * x^{3} * y^{3}}{5 * 6 * y^{2} * x^{5}} \]
• Then, cancel out similar factors present in both parts: - The "5" cancels because it's in both the numerator and denominator. - \(x^{3}\) in the numerator cancels with three \(x\)'s from \(x^{5}\) in the denominator. - \(y^{2}\) cancels in the denominator, leaving \(y\) in the numerator.

This leads to a simplified expression of \(\frac{y}{2x^{2}}\).
By efficiently using cancellation and recognizing common factors, you can significantly streamline algebraic expressions.