Problem 5 - Elementary Algebra

Question

Find each product. $$ (x+2)^{3} \cdot(x+2)^{5} $$

Step-by-Step Solution

Answer

Answer: The product of the given expressions is \((x+2)^8\).

Step 1: Identify the base and exponents

The base in our expression is \((x+2)\), and the exponents are \(3\) and \(5\).

Step 2: Apply the exponent property of multiplying same bases

According to the rule, \((a^m)(a^n) = a^{m+n}\). So, for our given expression: $$ (x+2)^3 \cdot (x+2)^5 = (x+2)^{(3+5)} $$

Step 3: Add the exponents

Add the exponents \(3\) and \(5\) together: $$ (x+2)^{(3+5)} = (x+2)^8 $$

Step 4: Write the final answer

The product of the given expressions is \((x+2)^8\).

Key Concepts

Exponent Properties

Understanding exponent properties, or the laws of exponents, is crucial when dealing with algebraic expressions involving powers. An exponent represents the number of times a base is multiplied by itself. Consider the expression \(a^n\), where \(a\) is the base, and \(n\) is the exponent indicating that the base \(a\) is used as a factor \(n\) times.

One fundamental property is the Product of Powers rule, stating that when multiplying two expressions with the same base, you can add the exponents. The mathematical representation of this rule is \(a^m \cdot a^n = a^{m+n}\). It's a shortcut that simplifies multiplication by eliminating the need to multiply the base by itself many times over.

For instance, in the expression from the exercise \( (x+2)^3 \cdot (x+2)^5 \), both terms have the same base \( (x+2) \). Therefore, applying the Product of Powers rule results in a single expression with the base \( (x+2) \) raised to the power of \( 3+5 \), which is \(8\), as shown in the step-by-step solution. This exponent property is a foundational concept in algebra that significantly streamlines the calculation process.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. Variables are symbols that represent unknown values and are often represented by letters such as \(x\), \(y\), or \(z\).

An important aspect of working with algebraic expressions is understanding how to properly manipulate them according to algebraic rules and properties. In the context of our example, the expression \( (x+2)^3 \cdot (x+2)^5 \) involves both a variable \(x\) and a constant \(2\), combined within a power structure. The whole expression denotes a product of two powers with identical bases.

Manipulating algebraic expressions efficiently is key to simplifying complex problems into more manageable forms. The exercise demonstrates the significance of recognizing patterns, such as repeating bases with exponents, and applying the corresponding algebraic properties to simplify the expression without expanding it entirely.

Simplifying Expressions

The process of simplifying expressions in algebra is about rewriting them in the most concise and clear manner without changing their value. This often includes combining like terms, reducing fractions, or using exponent rules to minimize the expression's complexity. Simplification makes calculations easier and helps in understanding the underlying structure of the problem.

In polynomial multiplication, like in our example, simplifying the expressions before multiplying can save a lot of work. Instead of expanding \( (x+2)^3 \cdot (x+2)^5 \) directly, which would be labor-intensive, we use exponent properties to simplify the expression first. By adding the exponents, we get \( (x+2)^8 \), a much simpler form.

To improve the consumability of this concept for students, it's beneficial to draw their attention to the simplification techniques at the beginning of the process and encourage the consistent application of algebraic rules to reduce complex expressions step by step.