Problem 66 - Elementary Algebra

Question

Find the quotient of \(\frac{x^{6} y^{8}}{x^{4} y^{3}}\).

Step-by-Step Solution

Answer

Answer: The quotient is \(x^{2} y^{5}\).

Step 1: Identify the terms to be divided

In this exercise, we have two terms in the form of \(\frac{x^{6} y^{8}}{x^{4} y^{3}}\). Here, we will divide the x terms and the y terms separately.

Step 2: Apply the exponent rule for division

We will now divide the x terms and the y terms separately by applying the exponent rule for division (subtracting the exponents). For the x terms, we have \(x^{6}\) and \(x^{4}\), and for the y terms, we have \(y^{8}\) and \(y^{3}\).

Step 3: Subtract the exponents for x terms

To divide the x terms, we subtract the exponents: \(x^{6} ÷ x^{4} = x^{6-4} = x^{2}\)

Step 4: Subtract the exponents for y terms

To divide the y terms, we subtract the exponents: \(y^{8} ÷ y^{3} = y^{8-3} = y^{5}\)

Step 5: Combine the results

Now, combine the results from steps 3 and 4 to find the complete quotient: $$\frac{x^{6} y^{8}}{x^{4} y^{3}} = x^{2} y^{5}$$

Key Concepts

Understanding Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols. It is used extensively to describe relationships and solve equations. In algebra, letters often represent numbers and are combined with arithmetic operations to form expressions and equations.
Understanding algebra is fundamental for solving various mathematical problems, including those that deal with exponentiation and polynomial division.
In the case of exponentiation, algebra helps us simplify expressions by using the properties of exponents to make calculations more manageable.

Division of Polynomials

Division of polynomials, such as \(\frac{x^{6} y^{8}}{x^{4} y^{3}}\), involves dividing expressions with several algebraic terms. The division process simplifies complex polynomial expressions. The goal is often to reduce the polynomial to its simplest form by canceling common factors between the numerator and the denominator.
In our exercise, we are tasked with dividing each part of the expression (the x and y terms) separately. We simplify the expression by breaking it into individual terms and applying the appropriate rules, such as the division of terms and subtraction of exponents.
Remember, the division of polynomials can be tackled step by step, ensuring each component is simplified correctly. This keeps larger problems manageable and easy to understand.

Mastering Exponent Rules

Exponent rules are essential in algebra, especially when working with expressions that involve division and multiplication of like bases. When dividing variables with exponents, such as \(x^{6}\) and \(x^{4}\), use the rule that states you should subtract the exponents: the formula is \(a^{m} \div a^{n} = a^{m-n}\).
This simple rule makes complex expressions easier to handle and leads to simplified results. For example, to divide \(x^{6}\) by \(x^{4}\), subtract the exponents (\(6-4\)) to get \(x^{2}\). Similarly, with \(y^{8}\) and \(y^{3}\), subtracting gives \(y^{5}\).
These exponent rules help to keep calculations straightforward and enable students to solve algebraic problems more confidently. Mastery of exponent rules is crucial for success in more advanced mathematics topics.