Problem 25 - Elementary and Intermediate Algebra

Question

Write each expression in an equivalent form using an exponent. $$ (x-y)(x-y) $$

Step-by-Step Solution

Answer

The expression \( (x-y)(x-y) \) can be written as \( (x-y)^2 \).

Step 1: Identify the Expression

The given expression is \( (x-y)(x-y) \).This is a product of two identical terms, suggesting the use of exponent rules.

Step 2: Apply the Law of Exponents

According to the law of exponents, a term multiplied by itself can be expressed with an exponent: \( a \cdot a = a^2 \).

Key Concepts

Exponent Rules

When dealing with algebraic expressions, exponent rules are essential tools. They help simplify repeated multiplication of the same base. For example, if you have a term like \( a \cdot a \), you can use exponent rules to express this as \( a^2 \). This means \( a \) is multiplied by itself one more time than the number indicated by the exponent.

There are a few basic rules you should know:

• Product of Powers: \( a^m \cdot a^n = a^{m+n} \)
• Power of a Power: \( (a^m)^n = a^{m \cdot n} \)
• Power of a Product: \( (ab)^m = a^m \cdot b^m \)

These rules make it easier to work with variables and numbers raised to powers. Understanding these can help in recognizing patterns and simplifying expressions efficiently.

Laws of Exponents

The laws of exponents are foundational principles in algebra, especially when working with mathematical expressions that involve powers.

These laws make the manipulation and simplification of expressions not only faster but more intuitive. The primary laws you should keep in mind include:

• Zero Exponent Rule: Any base except zero raised to the power of zero is always 1, i.e., \( a^0 = 1 \).
• Negative Exponent Rule: A negative exponent means you take the reciprocal of the base. For example, \( a^{-n} = \frac{1}{a^n} \).
• Quotient of Powers: To divide like bases with exponents, you subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \).

These laws are not just rules to memorize; they help resolve complex algebraic expressions and equations that might appear in various mathematical problems.

Mathematical Expressions

Mathematical expressions may appear intimidating, but they're essentially phrases made up of numbers, operations, and sometimes variables.

They can represent real-world situations or abstract concepts. These expressions are crucial in algebra because they form the basis for equations and functions.

• An expression could be as simple as \( 4 + 3 \) or as complex as \( 3x^2 - 4xy + y \).
• In every expression, identifying parts like coefficients, variables, exponents, and terms helps in simplifying or solving them.
• Remember, unlike equations, expressions do not have an equals sign. They merely convey value or represent a process.

Codifying expressions with exponents helps in organizing them neatly and expressing ideas succinctly, much like in the original exercise when we say \( (x-y)(x-y) \) can be neatly expressed as \( (x-y)^2 \). Mastering the art of working with mathematical expressions will pave the way for more advanced problem-solving capabilities.