Problem 14 - Algebra 2

Question

Simplify each expression. \(\frac{\frac{c^{3} d^{3}}{a}}{\frac{x c^{2} d}{a x^{2}}}\)

Step-by-Step Solution

Answer

The simplified expression is \(c d^{2}\).

Step 1: Recognize and Structure the Compound Fraction

The given expression is a compound fraction: \( \frac{\frac{c^{3} d^{3}}{a}}{\frac{x c^{2} d}{a x^{2}}} \). A compound fraction is a fraction where either the numerator, the denominator, or both, are themselves fractions. The objective is to simplify this into a single fraction.

Step 2: Rewrite the Fraction Using Division

Rewriting the compound fraction as a division problem helps us simplify it: \[ \frac{c^{3} d^{3}}{a} \div \frac{x c^{2} d}{a x^{2}} \]. The division of fractions can be expressed as the multiplication of the first fraction by the reciprocal of the second fraction.

Step 3: Multiply by the Reciprocal

Take the reciprocal of the second fraction and multiply: \[ \frac{c^{3} d^{3}}{a} \times \frac{a x^{2}}{x c^{2} d} \]. This means you flip the second fraction and change the division into multiplication.

Step 4: Simplify the Expression

Cancel common factors in the numerator and denominator. You can cancel \(a\) and \(a\) from both, \(c^{2}\) from \(c^{3}\) leaving \(c\), and \(d\) from \(d^{3}\) leaving \(d^{2}\), and \(x\) from \(x^{2}\) leaving \(x\): \[ \frac{c d^{2} x}{x} \].

Step 5: Final Simplification

Notice that the \(x\) in the numerator and denominator can be canceled, leaving \(c d^{2}\): \[ c d^{2} \].

Key Concepts

Compound Fractions

Compound fractions might seem complex at first glance, but breaking them down can make them much simpler. A compound fraction is a fraction in which either its numerator, denominator, or sometimes both are fractions themselves. Such fractions can appear daunting due to their nested structure. However, the key to simplifying them lies in expressing the compound fraction as a division operation.

• Think of a compound fraction as a fraction of fractions.
• To simplify, the primary goal is to transform the complex structure into a single-layer fraction.

Once you recognize the components, you can start transforming it, usually by performing division operations which ease down the fraction complexity.

Division of Fractions

Division of fractions can be simplified by transforming it into a multiplication problem. When you divide by a fraction, you are essentially multiplying by its reciprocal.

• Start by rewriting the division as a multiplication of the first fractionby the reciprocal of the second fraction.
• For instance, dividing by \(\frac{b}{d}\) is the same as multiplying by \(\frac{d}{b}\).

This approach greatly simplifies the calculation process of otherwise seemingly complex fractional divisions. It also lays the groundwork for identifying and canceling common factors.

Reciprocals

The reciprocal of a fraction is simply a flipped version of that fraction. In other words, if you have a fraction \(\frac{a}{b}\), its reciprocal would be \(\frac{b}{a}\).

• Finding reciprocals is essential when you want to perform division of fractions.
• Once the reciprocal is determined, multiply it with the first fraction to simplify the problem.

By using reciprocals, you convert the division into a multiplication problem, making it much easier to manage. This method significantly cuts down on the number of steps required to simplify compound fractions.

Canceling Common Factors

Canceling common factors is an essential step in simplifying fractions once you've set up a less complex expression. After converting a compound fractioninto a single fraction through multiplication by a reciprocal, search for commonfactors in the numerator and the denominator.

• Identify variables or numbers that appear in both the numerator and denominator and cancel them accordingly.
• This step helps reduce the complexity of the fraction into its simplest form, making it easier to interpret and work with.

For example, if you have \(\frac{c^{3} d^{3}}{a} \times \frac{a x^{2}}{x c^{2} d}\),by canceling similar terms like \(a\), \(c^{2}\), and \(d\), you simplify itfurther to \(c d^{2} x\), and eventually \(c d^{2}\).
This step ensures the fraction is expressed in the simplest possible way,free from any unnecessary complexity.