Problem 48 - College Algebra and Calculus: An Applied Approach

Question

Rewrite the expression with positive exponents and simplify.\((x+y)^{-5}(x+y)^{9}\)

Step-by-Step Solution

Answer

\((x + y)^{4}\)

Step 1: Identify the base and the exponents

In our expression \((x+y)^{-5}(x+y)^{9}\) we recognize that \((x+y)\) is our base and -5 and 9 are our exponents.

Step 2: Apply the rule for multiplying expressions with the same base

When multiplying expressions with the same base, we add the exponents. This gives us \((x + y)^{-5 + 9}\).

Step 3: Simplify the exponent

Simplify the exponent by performing the operation: -5 + 9 = 4. This gives us the final expression: \((x + y)^{4}\)

Key Concepts

Negative Exponents Rule

Understanding the negative exponents rule is crucial when dealing with algebraic expressions. Simply put, any base raised to a negative exponent is equal to the reciprocal of the base raised to the opposite positive exponent. For example, \( a^{-n} \) is the same as \( \frac{1}{a^n} \).

When faced with an expression like \( (x+y)^{-5} \), applying the negative exponents rule means we would rewrite the expression as \( \frac{1}{(x+y)^5} \), effectively 'flipping' the base to the denominator with a positive exponent. This rule helps to transform expressions with negative exponents into a form that is typically easier to work with and further simplify.

Exponent Operations

Working with exponent operations requires a good grasp of the rules that govern how to manipulate them. When you multiply expressions with the same base like \( (x+y)^{-5}(x+y)^9 \), you simply add the exponents. This is because each exponent indicates how many times the base is used as a factor. Combining the factors results in the same base being used \( -5 + 9 \) times, leading to \( (x+y)^{4} \).

It's important to remember that this rule only applies when the base is the same and when the operation is multiplication. Other exponent operations, such as dividing expressions with the same base, involve subtracting exponents, while raising a power to another power involves multiplying exponents. Familiarizing yourself with these rules is essential for solving algebraic problems that involve exponents.

Algebraic Simplification

Algebraic simplification is the process of reducing expressions to their simplest form. This process often involves applying a series of algebraic rules systematically. For example, when we simplify \( (x+y)^{4} \) after combining our exponents, we've reached a form that would require further expansion if we needed to express it as a polynomial.

Simplification may include expanding powers, combining like terms, or factoring. In our expression, if we were to expand \( (x+y)^{4} \), we would use the Binomial Theorem or FOIL method for smaller powers to multiply out the terms. The goal is always to reach the most concise expression possible, allowing for easier evaluation or further mathematical manipulation. Keeping expressions simple helps to identify patterns and solve equations more effectively.