Problem 16 - Algebra 2

Question

Simplify. Assume that no variable equals 0. $$ \frac{-y^{5} z^{7}}{y^{2} z^{5}} $$

Step-by-Step Solution

Answer

Simplified expression: \(-y^{3} z^{2}\).

Step 1: Identify the Common Base

First, identify the terms with the same bases in both the numerator and the denominator. The common bases in this expression are \(y\) and \(z\).

Step 2: Apply the Quotient Rule for Exponents

Use the quotient rule for exponents which states \( \frac{a^m}{a^n} = a^{m-n} \). For the base \(y\), subtract the exponent in the denominator, 2, from the exponent in the numerator, 5: \[ y^{5-2} = y^{3} \].For the base \(z\), subtract the exponent in the denominator, 5, from the exponent in the numerator, 7:\[ z^{7-5} = z^{2} \].

Step 3: Combine the Results

Substitute these simplified terms back into the expression:\( \frac{-y^{5} z^{7}}{y^{2} z^{5}} = -y^{3} z^{2} \).

Key Concepts

Quotient Rule for Exponents

The quotient rule for exponents is a fundamental concept in simplifying rational expressions.
It provides a way to handle powers when dividing terms that share the same base. The rule states that when you divide like bases, you subtract the exponent in the denominator from the exponent in the numerator.For a general expression with like bases, this rule is expressed as:

• \( \frac{a^m}{a^n} = a^{m-n} \)

In this formula:

• \(a\) is the common base
• \(m\) and \(n\) are the exponents

This method simplifies expressions, helping you handle tasks that would otherwise involve cumbersome calculations with large exponents.
Always remember to ensure the bases are indeed the same before applying this rule.

Common Bases

Finding common bases is your first step when simplifying fractions with exponents.
It helps streamline the process significantly. In our provided expression, both the numerator and denominator contain bases that are the same—namely \(y\) and \(z\).Here's why identifying common bases is essential:

• It allows you to apply the quotient rule of exponents effectively.
• It ensures calculations are straightforward by focusing only on the powers without changing the base.

Common bases make simplification possible because they allow you to subtract exponents accurately when applying the quotient rule for exponents.
In practical terms, always search for and circle or highlight common bases to make utilizations of simplifications intuitive and efficient.

Exponent Subtraction

Exponent subtraction is an easy process that forms the core of the quotient rule for exponents.
Once you've identified the common bases, subtract the exponent of the denominator from that of the numerator.This is illustrated in our example:

• For the base \(y\), subtract 2 (denominator) from 5 (numerator): \(y^{5-2} = y^{3}\)
• For the base \(z\), subtract 5 (denominator) from 7 (numerator): \(z^{7-5} = z^{2}\)

By following the straightforward rule of subtracting exponents, you efficiently reduce each part of the expression.
It is important to perform these steps separately for each base, ensuring clarity and correctness in determining the simplified expression. Exponent subtraction not only simplifies the expression but also aids in understanding the behavior of exponents in division scenarios.