Problem 64 - Algebra for College Students

Question

Find each quotient. $$\frac{-48 a^{3} b c^{5}}{-6 a^{2} c^{4}}$$

Step-by-Step Solution

Answer

The quotient is \(8abc\).

Step 1: Identify the Common Factors

We start by looking at the given expression that is a fraction: \( \frac{-48 a^{3} b c^{5}}{-6 a^{2} c^{4}} \). Our goal is to simplify this fraction by canceling out common factors in the numerator and the denominator.

Step 2: Simplify the Coefficients

Divide the numerical coefficients in the fraction. The coefficient in the numerator is \(-48\), and in the denominator, it is \(-6\). \(-48\) divided by \(-6\) equals \(8\). This simplifies the fraction to \(8\cdot \frac{a^{3} b c^{5}}{a^{2} c^{4}}\).

Step 3: Simplify the Powers of \(a\)

In the numerator, we have \(a^{3}\) and in the denominator \(a^{2}\). To simplify, subtract the powers: \(a^{3-2} = a^{1} = a\). This updates the fraction to \(8a\cdot \frac{b c^{5}}{c^{4}}\).

Step 4: Simplify the Powers of \(c\)

In the numerator, \(c^{5}\) appears and \(c^{4}\) in the denominator. Hence, the simplification becomes \(c^{5-4} = c^{1} = c\). The fraction is now \(8a b c\).

Step 5: Combine All Parts

The simplification process resulted in \(8abc\). There are no terms left that need simplification or canceling out.

Key Concepts

Polynomial Division

Understanding polynomial division can make your algebraic journey much smoother. In this particular exercise, we are asked to find the quotient of a fraction. The key is to approach it like any division problem by simplifying each part of the expression.
In polynomial division, focus on dividing both the coefficients and variables. You divide the coefficients by performing regular division: in this exercise,

• -48 is divided by -6 and that yields 8.

Next, handle the variables separately. Often, you will be working with variable terms that can be simplified using exponent rules. This helps in reducing the complexity of the polynomials in both the numerator and the denominator.
Combine these simplifications, and you will find reducing polynomials to their simplest form easier with practice.

Exponent Rules

Exponent rules are incredibly useful when simplifying algebraic expressions. They help us deal with terms that are raised to specific powers and make them simpler to compute. The most common rule applied here is the division of terms with exponents.
Here’s the rule:

• When dividing two powers with the same base, subtract the exponents: \( x^m / x^n = x^{m-n} \).

In our exercise, the simplification of terms with the variable \( a \) shows this rule in action:

• \( a^3 / a^2 = a^{3-2} = a^1 = a \).

The same applies to the \( c \) variable,

• where \( c^5 / c^4 = c^{5-4} = c^1 = c \).

By using exponent rules, particularly the subtraction associated with division, we simplify the expression significantly.

Fraction Simplification

Fraction simplification in algebra is similar to basic fractions: it's about reducing the expression to its simplest form. When looking at algebraic fractions, aim to cancel any common factors in both the numerator and the denominator.
In our exercise, after simplifying the coefficients and applying exponent rules to the variables, we find:

• the expression simplified to 8 from the coefficients.
• The 'a', 'b', and 'c' terms simplified from their respective exponents.

After these reductions, we are left with

• \( 8abc \),

which means there are no more factors to cancel. When simplifying fractions, systematic cancellation of identical factors in both parts results in a cleaner, more manageable expression. This is an essential skill in algebra that boosts both efficiency and accuracy.