Problem 68 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The quotient is \(-a^{2}b\).

Step 1: Simplify Constants and Sign

The expression given is \(-\frac{a^{4} b^{5} c}{a^{2} b^{4} c}\). Notice first the negative sign. The expression is negative because of the negative sign in the numerator. So the simplified expression will also result in a negative fraction.

Step 2: Simplify Each Variable Separately

Now we simplify the expression by dividing the powers of each variable respectively:- For \(a\): \(a^{4} \div a^{2} = a^{4-2} = a^{2}\).- For \(b\): \(b^{5} \div b^{4} = b^{5-4} = b^{1} = b\).- For \(c\): \(c \div c = c^{1-1} = c^{0} = 1\). This means \(c\) cancels out.

Step 3: Combine the Simplified Parts

Combine the simplified parts keeping the negative sign from Step 1:The expression simplifies to: \(-a^{2}b\).

Key Concepts

Simplifying Expressions

Simplifying expressions is the process of making a complex expression easier to understand and work with by reducing it to its simplest form. In algebra, this often involves combining like terms, reducing fractions, or eliminating unnecessary parts of an expression.
In the given exercise, we started with the expression \(-\frac{a^{4} b^{5} c}{a^{2} b^{4} c}\). The first step is to understand which parts can be reduced. For instance, recognizing like terms or shared factors between the numerator and denominator allows us to simplify the expression.
Here’s a simple approach:

• Look for common factors between the numerator and denominator.
• Divide these common factors out, which helps in reducing the expression.
• Simultaneously, simplify any constants or coefficients.

Remember, the end goal of simplifying is to make the expression easier to work with, which often aids in problem-solving or further calculations.

Exponent Rules

Understanding exponent rules is crucial in algebra, especially when working with expressions like the one in the exercise. Exponents allow us to represent repeated multiplication concisely. Here, let's focus on some essential rules:
- **Product of Powers Rule**: When multiplying two powers with the same base, add their exponents. For example, \(a^m \times a^n = a^{m+n}\).
- **Quotient of Powers Rule**: When dividing two powers with the same base, subtract their exponents: \(a^m \div a^n = a^{m-n}\).
In our problem, we applied the quotient of powers rule to each variable:

• For \(a\), \(a^4 \div a^2 = a^{4-2} = a^2\).
• For \(b\), \(b^5 \div b^4 = b^{5-4} = b\).
• For \(c\), \(c \div c = c^{1-1} = c^0 = 1\), effectively removing \(c\).

Understanding and applying these rules simplifies expressions and makes equations more manageable.

Negative Signs

Negative signs in algebraic expressions can often cause confusion, but they are straightforward once the basic rules are grasped. Negative signs simply indicate the opposite or the additive inverse of a number.
In the context of fractions, a negative sign can appear in the numerator, the denominator, or outside the fraction. Its position typically doesn’t change the sign of the entire expression.
In the exercise \(-\frac{a^{4} b^{5} c}{a^{2} b^{4} c}\), the negative sign is in front, indicating that the entire fraction is negative. When simplifying fractions:

• Keep track of the negative sign to ensure it is accurately represented in the simplified expression.
• If there's a negative sign in both the numerator and denominator, they cancel each other out, making the fraction positive.

By keeping these principles in mind, handling negative signs becomes a straightforward task in algebraic expressions.