Problem 71 - Algebra for College Students

Question

Find each quotient. $$\frac{14 a b^{3}}{-14 a b}$$

Step-by-Step Solution

Answer

The quotient is \(-b^2\).

Step 1: Simplify the Coefficients

Look at the coefficients in the fraction: 14 in the numerator and -14 in the denominator. When you divide 14 by -14, it simplifies to -1. So, we have a fraction of -1.

Step 2: Simplify the 'a' terms

We have an 'a' in both the numerator and the denominator. Since they are the same, they cancel each other out, leaving 1.

Step 3: Simplify the 'b' terms

In the numerator, we have \(b^3\) and in the denominator, we have \(b\). When dividing, we subtract exponents: \(b^3 \div b^1 = b^{3-1} = b^2\).

Step 4: Combine the Results

Combine the simplified coefficients and the results from the 'a' and 'b' terms. We get -1 \(b^2\). Therefore, the expression simplifies to \(-b^2\).

Key Concepts

Division of Monomials

When dividing monomials, you're essentially looking to simplify a fraction made up of simpler algebraic expressions. In the expression \(\frac{14ab^3}{-14ab}\), each part—coefficients, variables, and their exponents—needs to be evaluated independently.

• Coefficients: First, focus on the numeric parts, which in this case are 14 and -14. These are divided to yield -1.
• Variables: For the variable \(a\), since \(a^1\) is in both the numerator and the denominator (even if the exponent isn't visibly written, it's understood to be 1), they cancel each other out, leaving a factor of 1.
• Exponents: Lastly, for \(b\), remember the powers; you'll manage these by using exponent rules, which we’ll further explore next.

This results in combining simplified characteristics to reach a solution.

Exponent Rules

Exponent rules are key to simplifying expressions involving powers, especially when variables are multiplied or divided. When dividing variables like \(b^3\) and \(b^1\), the rule is to subtract the exponents:

• For instance: \(b^3\div b^1 = b^{3-1} = b^2\).

This subtraction stems from the basic exponent rule where \(a^m \div a^n = a^{m-n}\).

• In our original expression, this means simplifying the \(b\) terms by subtracting the smaller exponent in the denominator from the larger exponent in the numerator, resulting in \(b^2\).

Understanding these rules makes it easier to manage terms with powers, maintaining accuracy while simplifying algebraic fractions.

Canceling Terms

Canceling terms is a straightforward process in algebraic fractions, which significantly simplifies expressions. When you have the same base variable in both the numerator and the denominator, it cancels to 1 due to division.

• Example: Given an expression like \(\frac{a}{a}\), it simplifies to 1 since any number divided by itself is 1.

In our specific problem, both \(a\) terms in the expression \(\frac{14ab^3}{-14ab}\) cancel each other out.
The coefficients can similarly be reduced, meaning the numeric component simplifies first to -1. Then, after canceling identical variables, any leftover variables, such as the \(b^2\), reflect the remaining expression.
Remember: canceling functions made easy once you recognize the like terms or numbers are significantly similar in both parts of a fraction.