Problem 17 - Elementary Algebra

Question

Find each quotient $$ \frac{26 x^{4} y^{6} z^{2}}{13 x^{2} y^{2} z} $$

Step-by-Step Solution

Answer

Answer: The simplified quotient of the given expression is $2x^2y^4z$.

Step 1: Write down the given expression

The given expression is: $$ \frac{26 x^{4} y^{6} z^{2}}{13 x^{2} y^{2} z} $$

Step 2: Simplify numbers and common variables

We need to simplify the constants and the variables in the given fraction. For constants, we can divide 26 by 13 and for variables, we use the rule that same bases can be divided by subtracting the exponents. So we have: $$ \frac{26}{13} \cdot \frac{x^{4}}{x^{2}} \cdot \frac{y^{6}}{y^{2}} \cdot \frac{z^{2}}{z^{1}} $$

Step 3: Divide constants and variables

Now, we can simplify the constants and variables: 1. Divide constants: \(\frac{26}{13} = 2\) (since 26 is divisible by 13, the constant part is 2) 2. Divide the variables with the same base: - \(\frac{x^4}{x^2} = x^{(4-2)} = x^{2}\) - \(\frac{y^6}{y^2} = y^{(6-2)} = y^{4}\) - \(\frac{z^2}{z^1} = z^{(2-1)} = z^1 = z\)

Step 4: Combine the simplified constants and variables

Finally, let's combine the simplified constants and variables: $$ 2 \cdot x^2 \cdot y^4 \cdot z $$

Step 5: Write the final quotient

The simplified version of the given expression is: $$ 2x^2y^4z $$

Key Concepts

Quotients in Algebra

A quotient is the result of dividing one number or expression by another. In algebra, we often encounter quotients when simplifying fractions or rational expressions. Each component of the fraction, like the numerator and the denominator, represents a part of the division process.
Understanding quotients is crucial, as it helps us break down complex expressions into simpler terms.

In the given exercise, we're asked to find the quotient of two algebraic expressions. This involves dividing the entire numerator by the entire denominator.
When simplifying such expressions:

• Always start by dividing the coefficients, or the constant numbers, on the top and bottom.
• Next, handle each variable individually, particularly by assessing their exponents.
• This simplification process reduces complex expressions, making them easier to interpret and solve.

Understanding Exponents

Exponents are a way to express repeated multiplication of the same number. For instance, \( x^4 \) implies that \( x \) is multiplied by itself four times.
Using exponents makes equations neater and more manageable.
Among varied rules of exponents, the division rule is prominently used when simplifying complex expressions.
When dividing like bases, the exponents are subtracted. For example, \( \frac{x^4}{x^2} \) results in \( x^{(4-2)} = x^{2} \).
This subtraction rule is central to solving expressions involving exponents like in the exercise provided.
In algebraic quotients:

• The same base terms are lined up against each other.
• Their exponents are subtracted, effectively simplifying the expression.
• Remember that any variable with an exponent of 1 like \( z^1 \) is simply \( z \).

Variable Simplification

Variable simplification helps in reducing expressions to their simplest forms. This simplicity aids both calculation and comprehension. In algebra, each variable part needs the careful application of exponent rules.
To simplify variables:

• Find terms with the same base in the numerator and denominator.
• Use the exponent subtraction rule where applicable.
• Once simplified, rewrite the expression, ensuring no unnecessary terms remain.

In this task, after dividing the constants, we look at the bases \( x, y, \) and \( z \). Each undergoes the exponent rule to simplify them into its lowest form.
Completing this simplification correctly can turn a complex expression into something as simple as \( 2x^2y^4z \), which is much easier to handle or further operate with.