Problem 5 - College Algebra with Modeling and Visualization

Question

$$ \frac{5^{m}}{5^{n}}=______ $$

Step-by-Step Solution

Answer

\( 5^{m-n} \)

Step 1: Identify the Base and Exponents

In the expression \( \frac{5^m}{5^n} \), notice that the base for both the numerator and the denominator is 5, and the exponents are \(m\) and \(n\) respectively.

Step 2: Apply the Quotient Rule for Exponents

The quotient rule states: \( \frac{a^b}{a^c} = a^{b-c} \). Therefore, apply this rule to the given expression \( \frac{5^m}{5^n} \), which results in \( 5^{m-n} \).

Key Concepts

Quotient Rule for Exponents

When working with exponents, one essential rule to understand is the quotient rule. This rule is incredibly useful when you have a division of same-base exponential terms. In simple terms, it tells us how to divide powers with the same base.
When you have something like \( \frac{a^m}{a^n} \), according to the quotient rule, you can simplify it by subtracting the exponent in the denominator from the exponent in the numerator. This results in the expression \( a^{m-n} \).
Remember!

• The base \(a\) must be identical for the quotient rule to apply.
• You take the exponent from the numerator (top part) and subtract the exponent from the denominator (bottom part).
• Subtraction is key here: \( a^{m-n} \) means you're reducing the power by the exponent in the denominator.

Just keep in mind, this rule doesn't work if the bases are different, for example, \( \frac{a^m}{b^n} \). Here, further simplifications are usually not possible.

Mathematical Expressions

In mathematics, expressions are groups of symbols that represent a value or a relationship. These can include numbers, variables, and operations like addition or multiplication.
Expressions like \( \frac{5^m}{5^n} \) are common, where you're dealing with variables and constants raised to certain powers.
Key Components of Expressions:

• **Numbers and Constants**: Fixed values like 5 in our expression \( 5^m \).
• **Variables**: Symbols that represent unknown values, like \( m \) and \( n \), which can take on different values.
• **Operations**: These are things we do with numbers and variables, like add, multiply, or divide.

In algebra, we're often working to simplify expressions or to solve them by finding the values of variables that make the expression true. Knowing how to simplify expressions, using rules like the quotient rule for exponents, is an important skill.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions or equations to make them easier to work with or to find solutions. It is a foundational skill that helps us to explore relationships between different mathematical entities.
In our example, \( \frac{5^m}{5^n} \), we used algebraic manipulation to apply the quotient rule. This helped us rewrite the expression as \( 5^{m-n} \).
Why Algebraic Manipulation is Useful:

• It allows us to simplify complicated expressions, making calculations easier.
• It helps in uncovering underlying patterns or relationships between variables.
• It is crucial in solving equations or inequalities where you need to isolate the variable.

Sometimes, you'll need to perform multiple steps of manipulation, moving terms across an equation, factoring, expanding, or using various rules like distributive or associative laws. The goal is to systematically break down the expressions into simpler forms or solve them efficiently.