Problem 10 - Elementary Algebra

Question

Suppose that the letters \(x\) and \(y\) are each used to represent numbers. Use exponents to express the following product. \(x \cdot x \cdot x \cdot x \cdot x \cdot y \cdot y \cdot y\)

Step-by-Step Solution

Answer

Question: Simplify the expression \(x \cdot x \cdot x \cdot x \cdot x \cdot y \cdot y \cdot y\) using exponents. Answer: The simplified expression is \(x^5 \cdot y^3\).

Step 1: Count Occurrences of Each Variable

In the expression \(x \cdot x \cdot x \cdot x \cdot x \cdot y \cdot y \cdot y\), the variable \(x\) appears 5 times and the variable \(y\) appears 3 times.

Step 2: Express Using Exponents

Since \(x\) appears 5 times, it can be written as \(x^5\). Similarly, since \(y\) appears 3 times, it can be written as \(y^3\). Now, we can express the original product using exponents: \[x \cdot x \cdot x \cdot x \cdot x \cdot y \cdot y \cdot y = x^5 \cdot y^3\]

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation. These expressions form the foundation of algebra and help us represent real-world situations in mathematical terms. For instance, the given problem

Suppose that the letters \(x\) and \(y\) are each used to represent numbers. Use exponents to express the following product.
\(x \cdot x \cdot x \cdot x \cdot x \cdot y \cdot y \cdot y\)

shows us a repetitive multiplication, which is a common occurrence in algebraic expressions.

When variables are multiplied by themselves, the algebraic expression can be simplified hugely by using exponents. Simplification allows us to perform operations easier and reduces the complexity of problems. In the curriculum, learning to recognize and consolidate repeated multiplication of variables is a key step toward mastering algebra.

Multiplication of Variables

Multiplication of variables follows the same fundamental principles as the multiplication of numbers. When we have the same variable being multiplied several times, we can simplify the expression using an exponent.

For example, let's improve the given exercise by providing context on the operation:

In the expression \(x \cdot x \cdot x \cdot x \cdot x\), here \(x\) is multiplied by itself 5 times. Naturally, we can express this multiplication as \(x^5\). This notation is not only cleaner but also an essential simplification tool in algebra.

By using exponents, we are reducing the cognitive load necessary for understanding, especially when dealing with large expressions with multiple variables. As students become familiar with this property, they can tackle more complex algebraic problems with ease.

Exponentiation

Exponentiation is a mathematical operation that involves raising a base to the power of an exponent. The base is the number being multiplied, and the exponent tells us how many times to multiply the base by itself. In the case of our exercise,

By recognizing that \(x\) is multiplied by itself 5 times, we use the exponent 5 and write \(x^5\). Similarly, the variable \(y\) multiplied by itself 3 times can be expressed as \(y^3\).

Through exponentiation, we streamline the manipulation of algebraic expressions containing repeated multiplication.

This method isn't restricted to variables—it applies to numbers too. When teaching this concept, breaking down the steps is crucial. Start with the base concept of what an exponent represents, then show examples using variables. Exponentiation is a cornerstone of algebra that will recur in many forms—from quadratic equations to exponential growth models.