Problem 76 - Elementary Algebra Within Reach

Question

$$ \text { In Exercises 73-76, rewrite the product in exponential form. } $$ $$ (u-v) \cdot(u-v) \cdot 8 \cdot 8 \cdot 8 \cdot(u-v) $$

Step-by-Step Solution

Answer

\( (u - v)^3 \cdot 8^3 \)

Step 1: Identify the Bases

The bases in the given expression are \(u - v\) and \(8\).

Step 2: Count Frequency of the Bases

\((u - v)\) appears 3 times and \(8\) appears 3 times in the expression.

Step 3: Rewrite in Exponential Form

Rewrite the expression as \( (u - v)^3 \cdot 8^3 \).

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators (like addition and multiplication) that represent mathematical relationships. In the expression \((u-v) \cdot (u-v) \cdot 8 \cdot 8 \cdot 8 \cdot (u-v)\), we have both variables \(u\) and \(v\) engaged in a subtractive relationship. The expression

• Involves variables (\(u\) and \(v\)) and constants (like \(8\)).
• Has terms grouped through multiplication.
• Represents repeated multiplication, which is where exponents come into play.

By recognizing the form and structure of algebraic expressions, you can simplify or transform them using various algebraic techniques.

Mastering Multiplication in Algebra

In algebra, multiplication of terms is a fundamental operation that combines factors into a single expression. When we see something like \((u-v)\cdot (u-v)\), it means that we multiply the expression \((u-v)\) by itself, resulting in equivalent exponential notation. Think of multiplication as combining identical groups:

• Analyzing how often each group repeats helps in transforming repeated multiplication into exponential terms.
• This is a stepping stone to simplifying and solving algebraic expressions efficiently.

Breaking multiplication into steps allows us to handle complex expressions more easily, leading naturally to the concept of exponents.

Exploring Exponents

Exponents allow us to rewrite repeated multiplication in a simpler form. Instead of writing \((u-v)\cdot(u-v)\cdot(u-v)\), you write \((u-v)^3\). Here's how exponents make life easier:

• Base: The repeated factor, like \(u-v\) or \(8\).
• Exponent: The number showing how many times the base is used as a factor, such as \(3\) in \((u-v)^3\).

With exponents, the original problem \((u-v) \cdot (u-v) \cdot 8 \cdot 8 \cdot 8 \cdot (u-v)\) simplifies to \((u-v)^3 \cdot 8^3\). Recognizing repeated patterns and applying exponents helps in reducing complexity in expressions. This not only saves space but also allows clearer visualization of mathematical relationships.