Problem 46 - Elementary and Intermediate Algebra

Question

Use the quotient rule for exponents to simplify each expression. Write the results using exponents. $$ \frac{r^{8} s^{9}}{r s} $$

Step-by-Step Solution

Answer

The expression simplifies to \( r^7 s^8 \).

Step 1: Identify the Quotient Rule

The quotient rule for exponents states that when dividing two expressions with the same base, you subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \). Here, we will apply this rule separately to each variable.

Step 2: Simplify the Expression for \( r \)

First, identify the exponents of \( r \). In the numerator, \( r \) has an exponent of 8, and in the denominator, \( r \) has an exponent of 1. Apply the quotient rule: \( r^{8-1} = r^7 \).

Step 3: Simplify the Expression for \( s \)

Next, look at the exponents of \( s \). In the numerator, \( s \) has an exponent of 9, and in the denominator, it has an exponent of 1. Apply the quotient rule: \( s^{9-1} = s^8 \).

Step 4: Write the Simplified Expression

Put together the simplified results for both variables: \( r^7 \) and \( s^8 \). The simplified expression becomes \( r^7 s^8 \).

Key Concepts

Simplifying Expressions

Simplifying expressions is like organizing your room. You take a messy pile of terms and make everything neat by combining like terms and following certain rules.

• Start by identifying similar components to organize them effectively. For exponents, you'll want to focus on terms that have the same base.
• Simplify each part of the expression step by step.

Remember, simplifying expressions is about making them as clear and straightforward as possible. It's important to apply the correct rules so the expression is easier to work with. This makes it simpler to understand or use in further calculations.

Exponents Subtraction

The concept of exponents subtraction comes into play when you're dividing terms with the same base. It's like having multiple layers of a cake. Once you cut a piece, you subtract layers from the whole.
The key rule here is the quotient rule: \[\frac{a^m}{a^n} = a^{m-n} \]This means you subtract the exponent of the term in the denominator from the exponent of the term in the numerator.

• Identify the base terms that are the same.
• Look at their exponents and subtract the denominator's exponent from the numerator's exponent.

Exponents subtraction is a straightforward yet powerful tool in algebra to simplify complex expressions.

Algebra Basics

Understanding algebra is like learning a new language. You start with the basics, like using variables, coefficients, and understanding operations. Simple rules like the distribution property or handling exponents form the foundation of algebra.

• Variables like \( r \) and \( s \) stand in for numbers and have exponents indicating repeated multiplication.
• Operations such as multiplication or division between these variables need specific rules, like the quotient rule for exponents.

Algebra fundamentals help you manipulate mathematical expressions to solve problems efficiently. They are the stepping stones to tackle more complex math concepts.