Problem 18 - Elementary and Intermediate Algebra

Question

Simplify each complex fraction. See Example \(1 .\) $$ \frac{\frac{14}{15 m}}{\frac{21}{25 m^{6}}} $$

Step-by-Step Solution

Answer

The simplified form of the complex fraction is \(\frac{10}{9}m^5\).

Step 1: Setup the Complex Fraction

The given complex fraction is \(\frac{\frac{14}{15m}}{\frac{21}{25m^6}}\). The goal is to simplify this expression by dealing with the fractional terms in the numerator and the denominator first.

Step 2: Simplify the Fraction by Division

A complex fraction \(\frac{a/b}{c/d}\) can be simplified by multiplying \(\frac{a}{b}\) by the reciprocal of \(\frac{c}{d}\) which is \(\frac{d}{c}\). So, rewrite: \[\frac{\frac{14}{15m}}{\frac{21}{25m^6}} = \frac{14}{15m} \times \frac{25m^6}{21}\].

Step 3: Multiply the Fractions

Now multiply the numerators and the denominators: \[\frac{14 \times 25m^6}{15m \times 21}\]. Compute the product to get: \(\frac{350m^6}{315m}\).

Step 4: Simplify the Expression

To simplify \(\frac{350m^6}{315m}\), first simplify the coefficients: determine the greatest common divisor (GCD) of 350 and 315, which is 35. So, \[\frac{350}{315} = \frac{10}{9}\].

Step 5: Simplify the Variables

Next, simplify the powers of \(m\) by subtracting exponents according to the division rule for exponents \(m^{a}/m^{b} = m^{a-b}\). Thus, \(\frac{m^6}{m} = m^{6-1} = m^5\).

Step 6: Combine Coefficient and Variables

Combine the simplified coefficient and the variable terms back into one fraction: \(\frac{10}{9}m^5\). This is the simplified form of the original complex fraction.

Key Concepts

Complex Fractions

Complex fractions can seem intimidating at first, but breaking them down is manageable with the right steps. A complex fraction involves a fraction where either the numerator, the denominator, or both, are also fractions.
Imagine them like a fraction within a fraction—a bit like stacking teacups. Tackling them involves simplifying the smaller fractions first, enabling you to focus on a single, simpler expression.
To handle these complex structures, you typically aim to eliminate the fractional division. This usually means you convert the complex fraction into a more manageable expression by multiplying the numerator by the reciprocal of the denominator.

Fraction Simplification

Simplifying fractions is like tidying up a room: removing clutter to see the essentials. The process begins by analyzing the fractions presented in both the numerator and the denominator.
Each fraction should initially stand alone, allowing you to clear paths through each by eliminating common factors.
Simplification involves:

• Finding and dividing by the greatest common divisor (GCD) for the coefficients (numbers); it's crucial for trimming away excess parts.
• Reducing terms step-by-step until only the bare necessities remain in the simplified version of your expression.

Although tedious, each simplification step leaves the fraction in a reduced form that is easier to manage.

Division Rule for Exponents

When faced with variables and exponents within fractions, the division rule for exponents comes to the rescue! This rule helps to simplify expressions that might otherwise look complicated.
The rule states that when you divide expressions with the same base, you subtract the exponents:
Given the formula:

• For two exponents with the same base, such as $m^a\; ext\{\; and\; \}m^b$, the operation $m^a/m^b\; ext\{\; simplifies\; to\; \}m^\{a-b\}$.

This method streamlines equations, reducing high powers to lower, more manageable numbers.