Problem 133 - Algebra A Combined Function

Question

Simplify each expression. Assume that variables represent positive integers. $$ a^{4 m+1} \cdot a^{4} $$

Step-by-Step Solution

Answer

The expression simplifies to \( a^{4m+5} \).

Step 1: Identify the Expression

We are given the expression \( a^{4m+1} \cdot a^{4} \). Our task is to simplify it using the laws of exponents.

Step 2: Apply the Product of Powers Rule

The Product of Powers Rule states that when multiplying like bases, we add their exponents: \( a^m \cdot a^n = a^{m+n} \). Here, our base \( a \) is common, so we add the exponents: \( (4m + 1) + 4 \).

Step 3: Add the Exponents

Perform the addition of the exponents: \( (4m + 1) + 4 = 4m + 1 + 4 = 4m + 5 \).

Step 4: Write the Simplified Expression

Now that we have summed the exponents, we write the simplified expression as \( a^{4m+5} \).

Key Concepts

Simplifying Expressions

Simplifying expressions is a fundamental concept in algebra that involves reducing an expression to its simplest form. In this context, we're focusing on expressions with exponents. When you see a complex expression like \( a^{4m+1} \cdot a^4 \), the goal is to combine and condense it.

• Identify common bases: Look for the same variable or base in the expression.
• Apply rules of exponents: Use rules like the product of powers rule to simplify.
• Combine like terms: Add the exponents to simplify the expression further.

By doing this, expressions become easier to work with or further manipulate in algebraic equations.

Product of Powers Rule

The Product of Powers Rule is one of the essential laws of exponents. It states that when you multiply two expressions with the same base, you can simplify by adding the exponents.
For example, if you have \( a^m \cdot a^n \), then according to the Product of Powers Rule, you simplify it to \( a^{m+n} \). This rule makes it easier to handle expressions with exponents.
Consider our original exercise: \( a^{4m+1} \cdot a^4 \). Both expressions share the base \( a \), so we add the exponents:

• First exponent: \( 4m+1 \)
• Second exponent: \( 4 \)
• Sum of exponents: \( (4m+1) + 4 = 4m + 5 \)

This gives us the simplified expression \( a^{4m+5} \). This rule helps simplify complex calculations significantly.

Mathematical Notation

Mathematical notation is a system of symbols used to represent mathematical concepts and operations. It’s like a universal language that mathematicians use to communicate complex ideas concisely and accurately.
In terms of exponents, mathematical notation allows us to simplify how we write repeated multiplication. Instead of writing \( a \cdot a \cdot a \cdot a \), we use \( a^4 \). The notation tells us a lot:

• The base \( (a) \): the number or variable that is being multiplied.
• The exponent \( (4) \): how many times the base is used as a factor.

Understanding notation is crucial as it forms the basis of interpreting and solving algebraic problems effectively.

Variables in Algebra

Variables in algebra are symbols or letters that represent numbers or values. They are essential because they allow us to write general and abstract equations or expressions, which can then be solved for particular values.
In the problem we are discussing, \( m \) is a variable:

• It stands in for an integer, making our expression \( a^{4m+1} \cdot a^4 \) adaptable to different scenarios.
• Variables allow flexibility; they can take on different values, so the expression covers various cases.

Mastering variables is crucial for solving algebraic equations and understanding how changes in one variable can impact an entire mathematical equation or system.