91Ó°ÊÓ

Exponents are a shorthand way to express repeated multiplication of the same number. In basic algebra, understanding how to manipulate exponents is essential. For example, the expression \(x^{500}\) means "\(x\) multiplied by itself 500 times". A fundamental principle of exponents is the division rule. When you divide exponents with the same base, you subtract the exponents: \(x^{m}/x^{n} = x^{m-n}\). This property is key to simplifying expressions and solving problems in algebra. It's important to remember that any number to the power of zero (e.g., \(x^0\)) equals one, which often helps in simplifying complex expressions. Always handle exponent operations with care to avoid mistakes.

Simplification in algebra involves reducing expressions to their simplest forms. This process makes expressions easier to understand and solve. In this exercise, the expression \(\frac{x^{500}}{x^{499}} - 1\) is simplified using the basic exponent rule \(x^{m}/x^{n} = x^{m-n}\). So, dividing \(x^{500}\) by \(x^{499}\) results in \(x^{1}\), which is simply \(x\). Then, having \(\frac{x^{499}}{x^{499}}\) results in \(1\), making the whole expression \(x - 1\).
Simplification allows us to work with more manageable numbers and expressions, which is especially helpful when substituting values or further algebraic operations. Remember to look for common factors and use exponent rules to simplify expressions efficiently.

Substitution is a method used in algebra to replace variables with their corresponding numerical values. It is often the final step in an algebraic problem when solving for a specific numerical answer. Once an algebraic expression is simplified, substitution makes it easy to evaluate the expression with the given values.

• In this exercise, after simplifying to \(x - 1\), the variable \(x\) is substituted with its given value, 105.
• This gives us \(105 - 1\), resulting in 104.

Substitution is a powerful tool because it bridges abstract algebraic expressions with concrete numbers, allowing us to find precise solutions.