Problem 65 - Algebra for College Students

Question

Find each quotient. $$\frac{-18 x^{2} y^{2} z^{6}}{x y z^{2}}$$

Step-by-Step Solution

Answer

The quotient is \(-18xyz^4\).

Step 1: Identify the numerator and the denominator

The expression given is \( \frac{-18x^2y^2z^6}{xyz^2} \). Here, the numerator is \(-18x^2y^2z^6\) and the denominator is \(xyz^2\).

Step 2: Simplify the coefficients

The coefficient of the numerator is \(-18\). There is no numeric coefficient in the denominator. This means the coefficient part of the quotient remains \(-18\).

Step 3: Simplify the variable \(x\)

In the numerator, we have \(x^2\), and in the denominator we have \(x^1\). Applying the division of powers rule \(x^a / x^b = x^{a-b}\), we get \(x^{2-1} = x^1 = x\).

Step 4: Simplify the variable \(y\)

In the numerator, we have \(y^2\), and in the denominator, we have \(y^1\). Therefore, \(y^{2-1} = y^1 = y\).

Step 5: Simplify the variable \(z\)

In the numerator, \(z^6\) is present, and in the denominator, \(z^2\) is present. Thus, \(z^{6-2} = z^4\).

Step 6: Combine all simplified terms

After simplifying each part, the expression becomes \((-18) x y z^4\). Therefore, the quotient is \(-18xyz^4\).

Key Concepts

Division of Powers Rule

When simplifying algebraic fractions, the Division of Powers Rule is a fundamental concept. It states that when you divide like bases with exponents, you can simply subtract the exponents. For instance, if you have \( x^a / x^b \), you simplify to \( x^{a-b} \). This helps in reducing the complexity of expressions involving powers by decreasing the number of exponents.

• Imagine having two terms with the same base, but different powers.
• Instead of separately calculating each power, you can find a single term simply by subtracting the powers.
• This makes expressions more manageable and often reveals more direct solutions.

Understanding this rule simplifies many complex algebraic problems, making the math more intuitive.

Numerator and Denominator

In any fraction, algebraic or numerical, the numerator and denominator lay the foundation of what the fraction represents. The numerator is the top part, and the denominator is the bottom part. Identifying these correctly is key, especially in algebra where they contain variables and coefficients.

• In our given expression \( \frac{-18x^2y^2z^6}{xyz^2} \), \(-18x^2y^2z^6\) is the numerator.
• The denominator is \( xyz^2 \).
• Both parts need to be simplified for reducing the fraction completely.

Correctly identifying and isolating these components is the very first step before any simplification processes can be applied.

Variable Simplification

Variable simplification involves reducing expressions involving variables to their most basic terms. This involves applying the Division of Powers Rule by canceling out common variables in both the numerator and denominator.

• Consider the variables from our expression: \(x^2, y^2, z^6\) in the numerator and \(x, y, z^2\) in the denominator.
• For \(x\), subtract the exponents: \(x^{2-1} = x\).
• For \(y\), it's a similar process: \(y^{2-1} = y\).
• For \(z\), subtract \(z^{6-2}\) to get \(z^4\).

The goal is to cancel out terms where possible and simplify to more manageable expressions. This makes complex equations easier to manipulate and solve.

Algebraic Expression Simplification

Simplifying algebraic expressions means breaking down the expression to its simplest form. This involves combining and reducing terms using various rules of algebra. The goal is to make algebraic problems easier to interpret and solve.

• Firstly, identify all parts of the expression to understand how they combine.
• Apply simplification rules, like handling coefficients separately from variables.
• Use Division of Powers Rule extensively to make terms less cumbersome.
• In our example, the coefficient \(-18\) is paired with all the simplified variable terms \(x, y, z^4\).

Simplification yields the cleanest version possible, in our case resulting in \(-18xyz^4\), providing clarity and readiness for further calculations if needed.