Problem 83 - Elementary Algebra

Question

For the following problems, write the expressions using exponential notation. $$ (-8)(-8)(-8)(-8) x x x y y y y y $$

Step-by-Step Solution

Answer

Question: Write the expression $(-8)(-8)(-8)(-8)(x)(x)(x)(y)(y)(y)(y)(y)$ in exponential notation. Answer: $(-8)^4x^3y^5$

Step 1: Group Factors#

First, separate the negative numbers, the x's and the y's. $$ (-8)(-8)(-8)(-8) \cdot x \cdot x \cdot x \cdot y \cdot y \cdot y \cdot y \cdot y $$

Step 2: Write the bases as exponential expressions#

Write the product of the negative numbers, x's and y's as base raised to exponent. $$ (-8)^4 \cdot x^3 \cdot y^5 $$

Step 3: Final Answer#

The given expression in exponential notation is: $$ (-8)^4x^3y^5 $$

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are a fundamental part of mathematics, providing a way to describe patterns, relationships and unknown values using symbols and numbers. They consist of variables, coefficients, constants, and operators that form a mathematical phrase. In our example, the expression \((-8)(-8)(-8)(-8) \cdot x \cdot x \cdot x \cdot y \cdot y \cdot y \cdot y \cdot y\) can be seen as an algebraic expression containing variables \(x\) and \(y\) and a numerical part \((-8)(-8)(-8)(-8)\).

• Variables: These are the letters in the expression, representing unknown values. Here, \(x\) and \(y\) are variables.
• Coefficients: These are numbers multiplying the variables. In our exercise, \(-8\) serves as a coefficient.
• Constants: Numbers independent of variables. Although not present separately in this exercise, constants play a crucial role in many expressions.
• Operators: The symbols \(\cdot\) indicate multiplication between numbers and variables.

Breaking down and grouping like terms, such as the variables and the numerical parts, helps simplify and express complex algebraic expressions concisely using exponential notation.

The Role of Exponents

Exponents are a compact way to represent repeated multiplication. They consist of a base and an exponent, portraying how many times that base is multiplied by itself. For example, in the expression \((-8)^4\), \(-8\) is the base, and \(4\) is the exponent, indicating \((-8)\) is multiplied by itself four times.

• An important function of exponents is simplifying expressions, reducing the need for lengthy multiplication chains.
• When working with variables, exponents follow similar rules. For instance, \(x^3\) means \(x\) multiplied by itself three times, and \(y^5\) indicates \(y\) multiplied by itself five times.
• Understanding exponents allows you to grasp the power of exponential growth and decay, crucial concepts in both mathematics and everyday life, such as understanding compound interest in finance or population growth.

By using exponents, we can write long multiplication sequences succinctly, making it easier to manipulate and communicate mathematical ideas.

Navigating Mathematical Notation

Mathematical notation is a standardized set of symbols and conventions used to express mathematical ideas clearly and efficiently. It serves as a universal language in mathematics, helping avoid confusion and enabling complex ideas to be expressed simply.

In the context of our exercise, exponential notation is a key part of this language, simplifying not only how we write expressions but also how we perform operations on them. Here’s how it helps:

• Clarity: Using exponential notation, such as \((-8)^4x^3y^5\), reduces ambiguity. It is immediately clear how many times each quantity is multiplied.
• Consistency: Since these notational conventions are widely accepted, they ensure that mathematical communication remains consistent across different contexts and populations.
• Efficiency: Notation allows us to represent large or complex ideas in a concise form, making calculations easier and more effective.

Proper use of mathematical notation, especially understanding the role of exponents, is essential for tackling increasingly complex problems in algebra and other mathematical fields.