Problem 15 - Algebra 2

Question

Simplify each expression. \(\frac{\frac{2 y}{y^{2}-4}}{\frac{3}{y^{2}-4 y+4}}\)

Step-by-Step Solution

Answer

\( \frac{2y(y-2)}{3(y+2)} \)

Step 1: Identify the Expressions

The expression given is a fraction where both the numerator and the denominator are themselves fractions: \( \frac{\frac{2y}{y^2-4}}{\frac{3}{y^2-4y+4}} \). The numerator of the main fraction is \( \frac{2y}{y^2-4} \) and the denominator is \( \frac{3}{y^2-4y+4} \).

Step 2: Simplify the Denominators

Firstly, we simplify the denominators. Notice the expressions in the denominators: 1. \( y^2 - 4 \) is a difference of squares and can be factored as \((y-2)(y+2)\).2. \( y^2 - 4y + 4 \) is a perfect square trinomial and can be factored as \((y-2)^2\).

Step 3: Rewrite the Fraction

After factoring, rewrite the original expression:\[ \frac{\frac{2y}{(y-2)(y+2)}}{\frac{3}{(y-2)^2}} \]

Step 4: Simplify by Multiplying

To simplify the complex fraction \( \frac{\frac{a}{b}}{\frac{c}{d}} \), multiply by the reciprocal: \( \frac{a}{b} \times \frac{d}{c} \). Apply this to our expression:\[ \frac{2y}{(y-2)(y+2)} \times \frac{(y-2)^2}{3} \]

Step 5: Cancel Common Factors

Now, simplify by canceling common factors in the numerator and denominator. The factor \( (y-2) \) is present in both terms, so it cancels out:\[ \frac{2y}{(y+2)} \times \frac{(y-2)}{3} = \frac{2y(y-2)}{3(y+2)} \]

Step 6: Write the Final Simplified Expression

The expression is now simplified to a single fraction:\[ \frac{2y(y-2)}{3(y+2)} \] This is the final simplified form.

Key Concepts

Factoring Polynomials

Factoring polynomials is like breaking down numbers into their prime factors, but with variables. It's an essential skill in algebra that simplifies complex expressions. For instance, consider the polynomial denominator in the expression \[ y^2 - 4. \] This can be factored into \((y-2)(y+2)\), known as the difference of squares. The difference of squares is a special case where you have two terms, both perfect squares, separated by a subtraction sign. Another common type is the perfect square trinomial. This is seen in the second denominator, \( y^2 - 4y + 4 \), which factors to \((y-2)^2\). Recognizing these patterns helps transform complex expressions into simpler ones, laying the groundwork for easier manipulation.

Complex Fractions

A complex fraction has fractions within fractions, which can be intimidating at first glance. Simplifying them involves a few careful steps to prevent misunderstandings.The given expression, \( \frac{\frac{2y}{y^2-4}}{\frac{3}{y^2-4y+4}} \), requires us to tackle two fractions: one in the numerator and one in the denominator. The complexity is tamed by remembering one key step: multiply by the reciprocal. To decomplexify, turn the division of fractions into multiplication by flipping the denominator fraction: \( \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} \). Now, the expression is easier to handle and becomes: \[ \frac{2y}{(y-2)(y+2)} \times \frac{(y-2)^2}{3} \]. This step simplifies a daunting expression into something more approachable.

Simplifying Expressions

Once polynomials are factored and fractions are decomplexified, simplifying expressions becomes straightforward. We're often interested in reducing an expression to its simplest form, which means removing any common factors. In the expression \( \frac{2y}{(y+2)} \times \frac{(y-2)^2}{3} \), a shared factor \((y-2)\) can be canceled since it appears in both the numerator and the denominator. It's like simplifying numerical fractions: if 4 and 8 have a common factor of 2, you divide both by 2. Similarly, removing the \((y-2)\) leads to the simplified expression \[ \frac{2y(y-2)}{3(y+2)} \]. This ensures that we've made the expression as concise as possible, which makes it much easier for further algebraic operations or understanding its behavior.