Problem 1 - Elementary Algebra

Question

Write each of the following using exponents. \(a \cdot a \cdot a \cdot a\)

Step-by-Step Solution

Answer

Question: Rewrite the expression using exponents: \(a \cdot a \cdot a \cdot a\). Answer: \(a^4\)

Step 1: Count the number of times "a" is multiplied

In the expression \(a \cdot a \cdot a \cdot a\), the variable "a" is being multiplied four times.

Step 2: Write the expression using exponents

Since "a" is being multiplied four times, we can write the expression using exponents as \(a^4\). So, \(a \cdot a \cdot a \cdot a = a^4\).

Key Concepts

Exponent Notation

Exponent notation is a shorthand way of expressing repeated multiplication of the same factor. Instead of writing the same number or variable over and over, we use exponents to simplify our work. For example, when we have a variable, such as a, being multiplied by itself several times, like a multiplied by a four times, we can use exponent notation to write this in a more compact form.

The general form for exponent notation is base^exponent, where the base a is the number being multiplied, and the exponent is the number of times it appears as a factor in the multiplication. For our exercise, we write:
\[a \cdot a \cdot a \cdot a = a^4\]
This tells us that a is used as a factor four times. Exponents make algebraic expressions shorter and are crucial when dealing with large numbers or complex algebraic operations.

Multiplication of Variables

When multiplying variables in algebra, if the variables are the same, we use exponent notation to simplify the expression. Multiplying a variable by itself is called squaring if it's twice (e.g., \(a^2\)) or cubing if it's three times (e.g., \(a^3\)). Beyond that, we simply use the number of times the variable is multiplied as the exponent.

Here's a crucial tip:

• When multiplying variables, and only if they have the same base, we add the exponents. For example, \(a^2 \cdot a^3 = a^5\).

This is an essential rule in algebra, known as the 'Product of Powers' property. It's important to remember that this rule only applies to multiplication, not addition or subtraction of variables.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables (like a, b, x, or y), and arithmetic operations (like addition, subtraction, multiplication, and division). Exponents are also a part of these expressions. Writing expressions correctly is critical to solving algebra problems effectively.

For instance, turning a multiplication of variables into an exponent form simplifies the expression and reduces potential errors in further calculations. Consider the expression from the exercise. By turning \(a \cdot a \cdot a \cdot a\) into \(a^4\), we've efficiently transformed a lengthy multiplication into a compact exponent form. This skill is not just about writing; it will help you understand and perform algebraic operations more confidently.