Problem 60 - Algebra for College Students

Question

Find each quotient. $$\frac{12 x^{2} y^{7}}{6 x^{2} y^{3}}$$

Step-by-Step Solution

Answer

The quotient is \(2y^4\).

Step 1: Set up the Fraction

Write out the given fraction:\[\frac{12 x^{2} y^{7}}{6 x^{2} y^{3}}\]

Step 2: Simplify the Coefficient

Divide the coefficients (12 by 6):\[ \frac{12}{6} = 2\]

Step 3: Cancel Out Common Terms in Parts

Next, focus on the \(x^2\) terms: Since \(x^2\) is present in both the numerator and the denominator, you can cancel them out.

Step 4: Simplify the Variables (y)

Simplify the y-variables by using the properties of exponents: \[\frac{y^7}{y^3} = y^{7-3} = y^4\]

Step 5: Combine the Simplified Results

Combine all simplified parts to get the final answer:\[2y^4\]

Key Concepts

Simplifying Fractions

Simplifying fractions is the process of reducing them to their simplest form, where the numerator and the denominator have no common factors other than 1. This means finding the greatest common divisor (GCD) and dividing both the numerator and the denominator by this number.
Let's apply this to our problem. We start with the fraction \(\frac{12 x^{2} y^{7}}{6 x^{2} y^{3}}\). We first divide 12 by 6, the greatest common divisor of the coefficients, which reduces the fraction to \(2\).
This shows us how fractions can be broken down step-by-step, making complex fractions much simpler.

Properties of Exponents

Understanding exponents is essential in algebra, especially when simplifying algebraic fractions. Here, we're focusing on the rule that states: when dividing like bases with exponents, you subtract the exponents.
In our exercise, the focus surfaces when dealing with the variable \(y\). We have \(y^7\) in the numerator and \(y^3\) in the denominator. By applying the properties of exponents, you subtract 3 from 7, resulting in \(y^4\).

• When dividing powers of the same base, subtract the exponent of the denominator from that of the numerator.
• Always ensure the base stays the same during this operation.

This property efficiently simplifies expressions involving exponents.

Polynomial Division

Polynomial division involves dividing coefficients of like terms and managing the exponents accordingly. While not strictly polynomial division, the technique used in the exercise resembles the simpler aspect of this concept.
Here, we divided each term separately. We dealt with the coefficients \(12\) and \(6\) just like a basic divide and extended this principle to the variables \(x^2\) and \(y^7\).
Though polynomial division in higher degrees might involve long division or synthetic division, the fundamental concept remains to divide each term effectively, simplifying the polynomial expression step by step.

Rational Expressions

Rational expressions are like numerical fractions, but they feature polynomials in their numerators and denominators. Simplifying rational expressions follows the same principles as simplifying fractions:

• Find common factors in the numerator and denominator.
• Cancel them out to reduce the expression.

In our problem, \(x^2\) was present in both the numerator and denominator. This allowed us to cancel it outright, streamlining the rational expression.
Understanding this allows you to handle more complex expressions ranging through fraction-based equations or inequalities. Simplicity and elegance in handling such tasks come from these fundamental steps.