Problem 14 - Elementary Algebra

Question

For Problems \(1-24\), divide the monomials. $$ \frac{24 x^{3} y^{4}}{x^{2} y^{2}} $$

Step-by-Step Solution

Answer

The result is \( 24xy^2 \).

Step 1: Simplify the Coefficients

Start by dividing the coefficients in the numerator (24) and the denominator (implicitly 1, since it's not shown for the variables).Divide 24 by 1:\[ \frac{24}{1} = 24 \]

Step 2: Apply the Law of Exponents to Variables

Use the law of exponents for division: \( \frac{a^m}{a^n} = a^{m-n} \). We will apply this to each variable separately.For \(x\):\[ \frac{x^3}{x^2} = x^{3-2} = x^1 \]For \(y\):\[ \frac{y^4}{y^2} = y^{4-2} = y^2 \]

Step 3: Combine Results

Now, combine the results of the simplified coefficient and variables:\[ 24 \cdot x \cdot y^2 \]Thus, the simplified expression is:\[ 24xy^2 \]

Key Concepts

Law of Exponents

The Law of Exponents is a fundamental concept in algebra that makes working with powers easier and more systematic. When dividing two expressions with the same base, the law tells us to subtract the exponent in the denominator from the exponent in the numerator.
For example, given an expression like \( \frac{x^m}{x^n} \), we simply perform \( x^{m-n} \). This subtraction process simplifies expressions and helps us easily manage potentially confusing power calculations.
This law applies separately to each variable within a monomial.

• In the given problem, we first look at the variable \( x \): \( \frac{x^3}{x^2} \) becomes \( x^{3-2} = x^1 = x \).
• Similarly, for the variable \( y \): \( \frac{y^4}{y^2} \) simplifies to \( y^{4-2} = y^2 \).

Remember, the key point is the bases must match; only then can we subtract the exponents!

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves reducing them to their simplest or most manageable form without changing their value. It's about making expressions easier to work with and understand.
When simplifying an expression like \( \frac{24 x^3 y^4}{x^2 y^2} \), each component—coefficients and variables—are handled separately. This process follows systematically:

• Firstly, attend to numerical coefficients by direct division.
• Next, focus on each variable by applying the Law of Exponents.

The objective is to have a clearer, more succinct expression that is straightforward. For instance, reducing to \( 24xy^2 \) uses fewer terms, yet remains equivalent to the original expression. The power lies in not just reducing calculation complexity, but also in enhancing our comprehension and ability to communicate ideas effectively.

Coefficient Division

At the heart of simplifying monomials is the process of coefficient division. Coefficients are the numerical factors in terms of algebraic expressions, and simplifying these often requires elementary arithmetic.
In our example, \( 24 \) is the coefficient in the numerator and it needs to be divided by the coefficient in the denominator. Since the denominator lacks a numerical coefficient, we assume it to be \( 1 \).

• This simplifies to dividing \( 24 \) by \( 1 \), remaining \( 24 \).

By conducting this division first, a solid numerical foundation is established that can later be multiplied by the simplified variables.
Handling coefficients upfront helps lessen the complexity of dealing with other parts of an expression, ensuring that the end result is both manageable and accurate.