Problem 54 - Algebra A Combined Function

Question

Perform the indicated operations. $$ \frac{-2 x}{x^{3}-8 x} \div \frac{3 x}{x^{3}-8 x} $$

Step-by-Step Solution

Answer

The result of the operation is \(-\frac{2}{3}\).

Step 1: Identify the division operation

The expression \( \frac{-2x}{x^{3}-8x} \div \frac{3x}{x^{3}-8x} \) indicates a division of two fractions. In division of fractions, you multiply the first fraction by the reciprocal of the second fraction.

Step 2: Write the reciprocal of the second fraction

The reciprocal of the fraction \( \frac{3x}{x^{3}-8x} \) is \( \frac{x^{3}-8x}{3x} \).

Step 3: Multiply the first fraction by the reciprocal of the second fraction

Change the division into multiplication by writing \( \frac{-2x}{x^{3}-8x} \times \frac{x^{3}-8x}{3x} \).

Step 4: Simplify the expression

Now, calculate the product: \[\frac{-2x}{x^{3}-8x} \times \frac{x^{3}-8x}{3x} = \frac{-2x \cdot (x^{3} - 8x)}{(x^{3}-8x) \cdot 3x}\].Cancel the common factor \((x^{3} - 8x)\) from the numerator and the denominator, resulting in:\[\frac{-2x}{3x}\].

Step 5: Further simplify the resulting fraction

Since both terms in the fraction \(\frac{-2x}{3x}\) have a common factor of \(x\), cancel out \(x\) from the numerator and denominator:\[\frac{-2}{3}\].

Key Concepts

Reciprocal of a Fraction

Dividing fractions might seem tricky at first, but there's a simple trick: the reciprocal! When you hear "reciprocal," think of flipping a fraction upside down. Consider a fraction like \( \frac{a}{b} \). Its reciprocal is simply \( \frac{b}{a} \).

• The reciprocal of a number is what you multiply it by to get 1.
• If you start with a positive fraction, its reciprocal is also positive.
• If it's negative, the reciprocal keeps the same sign.

Understanding reciprocals can rescue you from the complexity of fraction division. Just flip the second fraction and change the division to multiplication. It's like magic but with numbers!
Using this method simplifies calculations by transforming the division problem \( \frac{-2x}{x^{3}-8x} \div \frac{3x}{x^{3}-8x} \) into a friendlier multiplication problem of \( \frac{-2x}{x^{3}-8x} \times \frac{x^{3}-8x}{3x} \). Once you flip, the hard part is over.

Simplifying Expressions

The next step is breaking down complex expressions into simpler forms. Simplifying expressions means reducing them to their most understandable form. For instance, in the expression \( \frac{-2x \cdot (x^{3} - 8x)}{(x^{3}-8x) \cdot 3x} \), we look for areas that can be reduced or canceled.

• First, identify and cancel out common factors in both the numerator and denominator.
• Then, perform any arithmetic by multiplying or dividing what's left.

By simplifying, we turn a tangled web into a neat package, making it easier to spot patterns or further simplify. Simplification helps with understanding how different parts of an expression relate to each other. In our exercise, this resulted in the simplified fraction \( \frac{-2}{3} \) after canceling out terms.

Common Factors

Finding common factors is key when simplifying expressions, especially fractions. A common factor is a number or expression that divides exactly into two or more numbers or expressions. When both the top and bottom of a fraction share a factor, you can simplify it.

• Look for terms, variables, or numbers that can be divided equally.
• Cancel them out from the numerator and the denominator.

In the given problem, this technique helps reduce complexity. With \( \frac{-2x \cdot (x^{3} - 8x)}{(x^{3}-8x) \cdot 3x} \), the common factor \((x^{3} - 8x)\) was canceled. This step is central to solving fraction problems efficiently. It unclutters your math work, showing just the essence of the fraction.

Multiplying Fractions

Once we've flipped the second fraction, we multiply fractions like pros. Multiplying fractions involves multiplying the numerators together and the denominators together.

• Keep in mind the order doesn't matter. This is commutative property at play.
• Focus on any common factors can simplify before you multiply.

The task \( \frac{-2x}{x^{3}-8x} \times \frac{x^{3}-8x}{3x} \) turns into multiplying \(-2x\) by \(x^{3} - 8x\), both found in the numerator, and \( x^{3}-8x \) by \(3x\), in the denominator. Thanks to common factors, this complicated expression becomes manageable. Multiply the remaining terms, and voila, you're closer to finishing! The product's already been simplified, leading us to the answer \( \frac{-2}{3} \).