Question

Perform the indicated operation and simplify the result. Leave your answer in factored form. $$ \frac{\frac{12 x}{5 x+20}}{\frac{4 x^{2}}{x^{2}-16}} $$

Short answer

\(\frac{3(x-4)}{5x}\)

Step-by-step solution

Simplify the Numerator

Simplify the expression in the numerator \(\frac{12x}{5x+20}\). Factor out any common factors in the denominator: \(\frac{12x}{5(x+4)}\).

Simplify the Denominator

Simplify the expression in the denominator \(\frac{4x^2}{x^2-16}\). Notice that the denominator is a difference of squares: \(x^2 - 16 = (x-4)(x+4)\). Hence, the expression becomes \(\frac{4x^2}{(x-4)(x+4)}\).

Form the Division of Fractions

Rewrite the original problem as a division of fractions: \(\frac{\frac{12x}{5(x+4)}}{\frac{4x^2}{(x-4)(x+4)}}\). Division of fractions is performed by multiplying the first fraction by the reciprocal of the second fraction.

Multiply by the Reciprocal

Multiply by the reciprocal: \(\frac{12x}{5(x+4)} \times \frac{(x-4)(x+4)}{4x^2}\).

Simplify the Expression

Cancel out the common factors in the numerators and denominators: \(\frac{12x \cdot (x-4) \cdot (x+4)}{5(x+4) \cdot 4x^2}\). The common factors \((x+4)\) and one \(x\) simplify to: \(\frac{12 (x-4)}{5 \cdot 4x} = \frac{3 (x-4)}{5x}\).

Key concepts

Fraction Simplification

Fraction simplification is a key concept in algebra that involves reducing fractions to their simplest form. This process often makes the equation easier to work with and understand. When simplifying algebraic fractions, we look for common factors in the numerator and the denominator.

For example, in our exercise, the numerator is \(\frac{12x}{5x+20}\). Here, we can factor the denominator: \(\frac{12x}{5(x+4)}\). This step is crucial as it allows us to cancel out any common terms in subsequent steps.

Simplifying fractions helps to avoid complex calculations and makes solving problems much more straightforward.

Factored Form

Factored form is an expression represented as a product of its factors. It is often used to simplify complicated algebraic expressions and make them more manageable.

In our exercise, we converted the denominator of the fraction \(\frac{4x^2}{x^2 - 16}\) to its factored form as \(\frac{4x^2}{(x-4)(x+4)}\). Recognizing patterns like the difference of squares \((x^2 - 16 = (x-4)(x+4))\) helps in breaking down complex problems.

Factored form plays a critical role in simplification, as it allows us to cancel out common factors. Remember, the factored form is not just useful for this type of problem—it’s a versatile tool in many areas of algebra.

Reciprocal

The reciprocal of a fraction is obtained by swapping its numerator and denominator. This concept is fundamental when dealing with division of fractions. To divide by a fraction, we multiply by its reciprocal.

In our exercise, when we rewrite the division as a multiplication, we use the reciprocal: \(\frac{\frac{12x}{5(x+4)}}{\frac{4x^2}{(x-4)(x+4)}} = \frac{12x}{5(x+4)} \times \frac{(x-4)(x+4)}{4x^2} \).

Multiplying by the reciprocal simplifies the expression and enables us to identify and cancel out common factors, simplifying the overall calculation.