In Exercises 93–102, factor and simplify each algebraic expression. (x+5)^−1/2−(x+5)^−3/2

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1. Rewrite the expression using a common base. Both terms involve the base (x + 5) raised to different exponents. Let’s rewrite the expression as: (x + 5)^(-1/2) - (x + 5)^(-3/2).
2. Factor out the smallest power of (x + 5) from both terms. The smallest power here is (x + 5)^(-3/2). Factoring this out gives: (x + 5)^(-3/2) * [(x + 5)^(1) - 1].
3. Simplify the term inside the brackets. The expression becomes: (x + 5)^(-3/2) * [(x + 5) - 1].
4. Combine the terms inside the brackets. Simplify (x + 5) - 1 to get x + 4. The expression now becomes: (x + 5)^(-3/2) * (x + 4).
5. Express the final result in simplified form. The factored and simplified expression is: (x + 4) / (x + 5)^(3/2).

Key Concepts

Exponents and Negative Exponents

Exponents represent repeated multiplication of a base number. A negative exponent indicates the reciprocal of the base raised to the absolute value of the exponent. For example, a^(-n) = 1/(a^n). Understanding how to manipulate negative exponents is crucial for simplifying expressions like (x+5)^(-1/2) and (x+5)^(-3/2).

Factoring Algebraic Expressions

Factoring involves rewriting an expression as a product of its factors. This process is essential for simplifying complex algebraic expressions. In the given expression, recognizing common factors can help in reducing the terms effectively, making it easier to simplify the overall expression.

Simplifying Algebraic Fractions

Simplifying algebraic fractions involves reducing the fraction to its simplest form by canceling common factors in the numerator and denominator. This process often requires factoring and understanding the properties of exponents. In the context of the given expression, simplifying will lead to a clearer and more manageable form.