Problem 45 - Prealgebra

Question

Reduce each fraction to lowest terms. $$\frac{102 a^{2} b c^{3}}{114 a b^{2} c^{2}}$$

Step-by-Step Solution

Answer

The simplified fraction is \( \frac{17ac}{19b} \).

Step 1: Simplify the Coefficients

First, let's simplify the coefficients. The numerator's coefficient is 102 and the denominator's coefficient is 114. Determine the greatest common divisor (GCD) of 102 and 114. \( 102 = 2 \times 3 \times 17 \) and \( 114 = 2 \times 3 \times 19 \). The GCD is \( 2 \times 3 = 6 \). Divide both coefficients by 6: \( \frac{102}{6} = 17 \) and \( \frac{114}{6} = 19 \). The simplified coefficient fraction is \( \frac{17}{19} \).

Step 2: Simplify the Variable \(a\)

Review the \(a\) terms in the fraction. In the numerator, the term is \(a^2\) and in the denominator, it is \(a^1\). Apply the rule for division of exponents: subtract the exponent in the denominator from the exponent in the numerator: \( a^{2-1} = a^1 \).

Step 3: Simplify the Variable \(b\)

Look at the \(b\) terms. In the numerator, it is \(b^1\) and in the denominator, \(b^2\). Subtract exponents for division: \( b^{1-2} = b^{-1} \). The negative exponent can be rewritten as \( \frac{1}{b} \), moving \(b\) from the numerator to the denominator as \(b^1\).

Step 4: Simplify the Variable \(c\)

Consider the \(c\) terms. The numerator has \(c^3\) and the denominator has \(c^2\). Subtract exponents: \( c^{3-2} = c^1 \).

Step 5: Write the Fraction in Simplified Form

Combine the simplified coefficients and variables. The simplified fraction is \( \frac{17a c}{19b} \).

Key Concepts

Greatest Common Divisor (GCD)

When simplifying fractions, one of the key components is finding the Greatest Common Divisor (GCD). The GCD is the largest whole number that can divide both the numerator and the denominator without leaving a remainder. This step is crucial in reducing fractions to their simplest form.
To find the GCD, follow these steps:

• List the prime factors of each number. For example, if you have 102, you break it down into prime factors to get \( 2 \times 3 \times 17 \).
• Do the same for 114, which breaks down into \( 2 \times 3 \times 19 \).
• Identify the common prime factors between the numerator and denominator. In this case, both numbers share \( 2 \) and \( 3 \).
• Multiply the common factors to get the GCD. Here, it is \( 2 \times 3 = 6 \).

By dividing the original coefficients by their GCD, you simplify the fraction's numerical value efficiently.

Exponents and Powers

Exponents are a shorthand way of expressing repeated multiplication of a number by itself. In fraction simplification, understanding how to manage exponents is important when both the numerator and denominator contain powers of the same base.
When dividing terms with the same base, use the rule of subtracting exponents:

• Consider \( a^2 \) over \( a^1 \). Subtract the exponents: \( a^{2-1} = a^1 \) or simply \( a \).
• When the numerator has a smaller exponent, such as \( b^1 \) over \( b^2 \), the result is \( b^{1-2} = b^{-1} \).
• A negative exponent indicates the reciprocal. So \( b^{-1} = \frac{1}{b} \).

Through these principles, you're equipped to systematically simplify any expression involving exponents in fractions.

Fraction Simplification

Reducing fractions involves a systematic approach to ensure all components of the fraction are expressed in their simplest form. With both numerical coefficients and algebraic terms:

• Start by simplifying the numerical part using the GCD, as demonstrated with \( \frac{102}{114} \) turning into \( \frac{17}{19} \).
• Next, handle the variables. For example, transform \( \frac{a^2}{a^1} \) into \( a \), \( \frac{b^1}{b^2} \) into \( \frac{1}{b} \), and \( \frac{c^3}{c^2} \) into \( c \).
• Finally, combine all simplified parts into a single expression: \( \frac{17ac}{19b} \).

By simplifying each part of a fraction—numerical and variable--you can ensure the expression is in its most reduced form, making it easier to understand and work with in further mathematical operations.