Problem 88 - Elementary Algebra

Question

Express \(x x x y y y y(a+b)(a+b)\) using exponents.

Step-by-Step Solution

Answer

Answer: \(x^3 y^4 (a+b)^2\).

Step 1: Organize similar terms together

We start by organizing the expression so that similar terms are together: \(x \cdot x \cdot x \cdot y \cdot y \cdot y \cdot y \cdot (a+b) \cdot (a+b)\).

Step 2: Count the occurrences of similar terms and use exponents

Now count how many times each term appears and write the expression using exponents: \(x^3 \cdot y^4 \cdot (a+b)^2\). So, the expression \(x x x y y y y(a+b)(a+b)\) can be rewritten as \(x^3 y^4 (a+b)^2\) using exponents.

Key Concepts

Understanding Algebra

Algebra is a branch of mathematics that uses symbols and letters to represent numbers and quantities in formulas and equations. It's like a universal language that helps us express mathematical ideas in a generalized way.
Instead of using specific numbers all the time, algebra allows us to work with unknowns and variables, such as \(x\), \(y\), or \(a\), \(b\). This helps in solving problems where certain values are not known initially.

• Variables: These are symbols that represent unknown numbers. In our exercise, \(x\) and \(y\) are variables.
• Operations: Basic operations such as addition, subtraction, multiplication, and division are used within algebraic expressions.

By arranging and rearranging these variables and operations, we can solve equations or simplify expressions to find possible values for these unknowns. This fundamental understanding of algebra sets the stage for exploring more complex mathematical concepts, like polynomials.

Exploring Polynomials

Polynomials are algebraic expressions that consist of variables and coefficients, structured together using addition, subtraction, and multiplication (but not division by a variable).
Each term in a polynomial is a product of a number, called a coefficient, and a variable raised to a non-negative integer power, called an exponent.

• Terms: In the given exercise, the terms include \(x^3\) and \(y^4\); these are simple polynomials.
• Expressions: A polynomial can have one or many terms. If an expression has one term, it's called a monomial, two terms make it a binomial, and three terms form a trinomial.
• Degree: The degree of a term is the exponent of the variable in the term. For example, the term \(x^3\) has a degree of 3.

Polynomials form the building block of various algebraic structures and are used in a wide range of applications, from basic arithmetic to complex calculus.

Simplification Techniques

Simplification is the process of reducing an expression to its simplest form, making it easier to handle or solve. In algebra, this often involves using exponents to express repetitive multiplication more succinctly.
Let's break down the simplification process for the given expression:

• Grouping similar terms: First, identify terms that are alike, such as the repeated multiplications of \(x\) and \(y\).
• Using exponents: Instead of writing \(x \cdot x \cdot x\), we can use \(x^3\). This is similar for \(y\), which becomes \(y^4\).
• Factorization: Lastly, expressions like \((a+b)(a+b)\) can be written as \((a+b)^2\).

This method not only makes the expression cleaner but also helps in performing calculations faster and more efficiently. Recognizing patterns and applying exponents appropriately is key to mastering simplification.