Problem 3 - Elementary Algebra

Question

For exercises 1-66, simplify. $$ \frac{48 a^{2} b^{3}}{56 a b} $$

Step-by-Step Solution

Answer

\( \frac{6 a b^{2}}{7} \)

Step 1: Identify the common factors in the numerator and the denominator

The given fraction is \( \frac{48 a^{2} b^{3}}{56 a b} \). Identify the common factors in both the numerator and the denominator. Here, both the numerator and the denominator have common factors such as numbers and variables.

Step 2: Simplify the numerical coefficients

Divide the numerical coefficients 48 and 56 by their greatest common divisor (GCD). The GCD of 48 and 56 is 8. So, \( \frac{48}{56} = \frac{48 \div 8}{56 \div 8} = \frac{6}{7} \).

Step 3: Simplify the variable components

Simplify the variables in the fraction. The fraction becomes \( \frac{6 a^{2} b^{3}}{7 a b} \). Cancel out the common terms: one \(a \) and one \( b \) from both numerator and denominator, leaving \( a \) and \( b^{2} \) in the numerator.

Step 4: Combine results to get the simplified fraction

Combine the simplified numerical coefficient with the remaining variables. Thus, \( \frac{6 a^{2} b^{3}}{7 a b} \) simplifies to \( \frac{6 a b^{2}}{7} \).

Key Concepts

Numerator and Denominator

In algebraic fractions, the terms above and below the fraction line are referred to as the numerator and denominator, respectively. The numerator, in our example, is the expression on top: 48a²b³. The denominator is the expression on the bottom: 56ab. Each part of the fraction holds variables and numerical coefficients.
The first step in simplifying is to identify these elements. By separating the variables from the numerical coefficients, simplification becomes easier.

• Numerator: 48a²b³.
• Denominator: 56ab.

The key is to look for common factors between the numerator and the denominator, including both numerical coefficients and variables.

Greatest Common Divisor (GCD)

The greatest common divisor (GCD) is a fundamental concept in simplifying algebraic fractions. It refers to the largest number that can evenly divide multiple numbers. In our example, you need the GCD of 48 and 56. Determine the GCD to simplify numerical coefficients.
To find the GCD:

• List the factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.
• List the factors of 56: 1, 2, 4, 7, 8, 14, 28, 56.

The largest common factor is 8. Divide both 48 and 56 by this GCD:

• 48 ÷ 8 = 6.
• 56 ÷ 8 = 7.

Now, the numerical part of the fraction becomes \(\frac{6}{7}\). This method reduces larger numbers and makes the fraction simpler.

Simplifying Variables

Simplifying variables in algebraic fractions involves canceling common terms. In the expression \( \frac{6 a^2 b^3}{7 ab} \), we notice common variables in both the numerator and the denominator.
Identify the common variables:

• Numerator: a² and b³.
• Denominator: a and b.
• Common variable: one a and one b.

Cancel these common variables:

• a² ÷ a = a.
• b³ ÷ b = b².

Thus, the simplified variable part is reduced to ab².
Combine this with the simplified numerical coefficient to get the final simplification: \( \frac{6ab^2}{7} \). Breaking down each variable step by step ensures clarity and accuracy.