Problem 24 - Elementary Algebra

Question

For the following problems, write each of the quantities using exponential notation. \(2 \cdot 2 \cdot 5 \cdot 6 \cdot 6 \cdot 6 x y y z z z w w w w\)

Step-by-Step Solution

Answer

Question: Rewrite the given expression using exponential notation: \(2 \cdot 2, 5, 6 \cdot 6 \cdot 6, x, y \cdot y, z \cdot z \cdot z, w \cdot w \cdot w \cdot w\) Answer: \(2^2 \cdot 5 \cdot 6^3 \cdot x \cdot y^2 \cdot z^3 \cdot w^4\)

Step 1: Identify the numbers and variables with their repetition counts

We need to first find the repeated numbers and variables in the expression and count the number of times they are repeated. \(2 \cdot 2, 5, 6 \cdot 6 \cdot 6, x, y \cdot y, z \cdot z \cdot z, w \cdot w \cdot w \cdot w\)

Step 2: Write the expression in exponential notation

Now we rewrite the expression using exponential notation: \(2^2 \cdot 5 \cdot 6^3 \cdot x \cdot y^2 \cdot z^3 \cdot w^4\) So the given expression written in exponential notation is: \(2^2 \cdot 5 \cdot 6^3 \cdot x \cdot y^2 \cdot z^3 \cdot w^4\).

Key Concepts

Algebraic Expressions

Algebraic expressions are the backbone of algebra and are used to represent real-world and mathematical problems. They comprise numbers, variables, and operations (such as addition, subtraction, multiplication, and division). For example, look at the expression given in the exercise, \(2 \cdot 2 \cdot 5 \cdot 6 \cdot 6 \cdot 6 x y y z z z w w w w\). It contains numbers (2, 5, 6), variables (\(x, y, z, w\)), and multiplication operations. By understanding how to combine these elements, we can simplify the expression to make it more manageable and understandable.

Exponents

Exponents are a shorthand way to express repeated multiplication of the same number, known as the base. In algebra, using exponents helps to keep expressions succinct and reduces the complexity of calculations. The number of times the base is multiplied by itself is called the exponent or power. In the expression \(2^2 \cdot 5 \cdot 6^3 \cdot x \cdot y^2 \cdot z^3 \cdot w^4\), the exponent `2` for the base `2` tells us that `2` is multiplied by itself two times. Similarly, `6` raised to the third power means we multiply `6` by itself three times. Exponents can greatly simplify the calculation and interpretation of algebraic expressions.

Mathematical Notation

Mathematical notation is a system of symbols used to represent numbers, operations, and relationships in mathematics. It is universally understood, making it easier for students and mathematicians from around the world to communicate complex ideas concisely and precisely. For instance, exponential notation is a form of mathematical notation used to simplify expressions, especially when dealing with large numbers or many repetitions of the same number or variable. In the step-by-step solution, we replaced \(6 \cdot 6 \cdot 6\) with \(6^3\), thus using fewer symbols to convey the same meaning. This concise representation is both a convenience and a necessity for clarity in mathematical communication.