Problem 15 - Algebra A Combined Function

Question

Simplify each complex fraction. $$ \frac{-\frac{5}{12 x^{2}}}{\frac{25}{16 x^{3}}} $$

Step-by-Step Solution

Answer

The simplified fraction is \( \frac{-4x}{15} \).

Step 1: Understand the Problem

The given complex fraction is \( \frac{-\frac{5}{12 x^{2}}}{\frac{25}{16 x^{3}}} \). We need to simplify this fraction into a simpler form.

Step 2: Rewrite as Division

A complex fraction is essentially a division of two fractions. Rewrite the fraction as a division: \[ -\frac{5}{12x^2} \div \frac{25}{16x^3} \].

Step 3: Division of Fractions

To divide by a fraction, multiply by its reciprocal. The reciprocal of \( \frac{25}{16x^3} \) is \( \frac{16x^3}{25} \). Apply this to our expression.\[ -\frac{5}{12x^2} \times \frac{16x^3}{25} \].

Step 4: Multiply the Fractions

Multiply the numerators and the denominators:- Numerator: \( -5 \times 16x^3 = -80x^3 \).- Denominator: \( 12x^2 \times 25 = 300x^2 \).Thus, the expression becomes \( \frac{-80x^3}{300x^2} \).

Step 5: Simplify the Resulting Fraction

First, simplify the coefficient \( \frac{-80}{300} \). The greatest common divisor of 80 and 300 is 20, so divide both by 20: \[ \frac{-80}{300} = \frac{-4}{15} \].

Step 6: Simplify the Variable Part

Simplify the variable part by reducing the powers: - In the numerator, \( x^3 \), and in the denominator, \( x^2 \).- So \( \frac{x^3}{x^2} = x^{3-2} = x \).Hence, the simplified expression is \( \frac{-4x}{15} \).

Step 7: Present the Final Simplified Form

Combine both simplifications from Steps 5 and 6: \[ \frac{-4x}{15} \]. This is the simplest form of the original expression.

Key Concepts

Simplifying Fractions

Simplifying fractions involves reducing the fraction to its simplest form. This means breaking down the numerator and the denominator by their greatest common divisor (GCD), which is the largest number that evenly divides both the numerator and the denominator.
For any fraction, start by examining the numbers to see what they have in common. Reducing each side by this common number will help simplify the fraction.

• Identify the greatest common divisor of the numerator and the denominator.
• Divide both the numerator and the denominator by this number.
• The result is a fraction in its simplest form.

Doing this makes computations simpler and fractions easier to understand.

Division of Fractions

Dividing fractions is all about converting the division problem into a multiplication problem. This is accomplished by using the reciprocal of the divisor fraction.
Fraction division can seem tricky, but it's a straightforward process once broken down:

• Convert the division sign into a multiplication sign.
• Flip the second fraction (this is now the reciprocal).
• Multiply the fractions directly by multiplying their numerators and denominators.

This method makes it easier to handle complex fractions or when dividing any two fractions. It keeps everything systematic and manageable.

Reciprocal of Fractions

Reciprocals are key when dealing with the division of fractions. The reciprocal of a fraction \( \frac{a}{b} \) is simply \( \frac{b}{a} \). This means swapping the numerator and the denominator.
Reciprocals are helpful for multiple reasons:

• They turn division problems into multiplication ones.
• Using reciprocals simplifies the arithmetic steps needed to solve an equation.
• This can be particularly useful when dealing with algebraic expressions, where flipping and multiplying can clarify complex operations.

Understanding the idea of a reciprocal helps streamline the process of solving complex fraction problems.

Greatest Common Divisor

The Greatest Common Divisor (GCD) is fundamental when simplifying fractions and expressions. It helps break down numbers to their simplest form by identifying the largest number that divides two given numbers without a remainder.
Finding the GCD is beneficial because:

• It reduces fractions efficiently, making them simpler.
• It applies to breaking down algebraic terms with numerical coefficients.
• Using the GCD reduces the complexity in fraction or expression form by eliminating common factors.

Understanding and effectively using the GCD simplifies problems, leading to quicker and clearer solutions.