Problem 74 - Algebra for College Students

Question

Find each quotient. $$\frac{-48 x y z^{2}}{2 x z}$$

Step-by-Step Solution

Answer

The quotient is \(-24yz\).

Step 1: Identify the Given Terms

We are given the expression \( \frac{-48xyz^2}{2xz} \). Our goal is to simplify this expression to find the quotient.

Step 2: Simplify the Coefficients

The coefficients of the terms are \(-48\) and \(2\). Simplify this by performing the division \( \frac{-48}{2} = -24 \).

Step 3: Simplify the Variables

Look at the variable terms in the numerator and denominator. In \(xyz^2\), the terms \(x\) and \(z\) in the denominator \(xz\) can cancel out with the ones in the numerator. We are left with \(y\) in the numerator and \(z^1\) after cancelling one \(z\).

Step 4: Write the Simplified Expression

After simplifying both the coefficients and the variables, we are left with \(-24yz\). So, the quotient is \(-24yz\).

Key Concepts

Quotient

In mathematics, a quotient is the result you get when you divide one quantity by another. This can be as simple as dividing whole numbers, or as complex as simplifying algebraic fractions. In algebra, when you see an expression like \( \frac{a}{b} \), you are looking for the quotient resulting from the division of \( a \) by \( b \). Each part, the numerator (top) and the denominator (bottom), can have numbers (coefficients) and variables, or just one of them.
With algebraic expressions, finding the quotient involves simplifying these coefficients and variables.
Getting to the simplest form requires reducing both the numbers and any common factors in the variables.

Variable Cancellation

In algebraic expressions, variables can sometimes be "cancelled" to simplify the problem. This process involves removing variables that appear in both the numerator and the denominator.
For example, consider the expression \( \frac{3x^2y}{xy} \). You can cancel one \( x \) from each \( x^2 \) in the numerator and \( x \) in the denominator. Simplifying further, you remove the common \( y \) as well. This cancellation leaves you with \( 3x \).

• Cancel common factors to reduce expressions.
• Remember: you can only cancel factors (those connected by multiplication).

Through variable cancellation, the expression is cleaner and easier to understand.

Coefficient Simplification

When simplifying algebraic expressions, reducing coefficients is crucial. Coefficients are the numerical part of a term, appearing in front of variables.
If you have an expression like \( \frac{-48}{2} \), simplifying involves performing basic arithmetic—dividing both coefficients to get \(-24\). It’s really about performing the division when you spot a number in both the numerator and denominator.

• Identify the numerical coefficients.
• Use division to simplify them.

This practice not only makes expressions more concise but also easier to work with in more complex problems.

Algebraic Fractions

Algebraic fractions, like regular fractions, have a numerator and a denominator. However, the difference is they contain variables within these parts. Dealing with algebraic fractions involves simplifying them just as you would with number fractions, incorporating steps like coefficient simplification and variable cancellation.
You need to ensure that:

• Variables and coefficients are simplified as much as possible.
• The expression is left in its simplest form, particularly important when solving equations.

When you master simplifying these algebraic fractions, it opens the door to solving more challenging algebraic tasks with greater confidence and precision.