Problem 84 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Simplify the expression. $$ \frac{\frac{1}{2} x^{-1 / 2}(x+y)^{1 / 2}-\frac{1}{2} x^{1 / 2}(x+y)^{-1 / 2}}{x+y} $$

Step-by-Step Solution

Answer

The simplified form of the expression is \(\frac{\sqrt{x+y}-\sqrt{x}}{2\sqrt{x}\sqrt{x+y}(x+y)}\).

Step 1: Identify the terms and exponents

In the expression, there are two terms separated by a subtraction sign, both divided by the factor (x+y), which are: Term 1: \(\frac{1}{2}x^{-1/2}(x+y)^{1/2}\) Term 2: \(\frac{1}{2}x^{1/2}(x+y)^{-1/2}\)

Step 2: Simplify each term using rules of exponents

In this step, we will simplify each term using the rules of exponents. Recall that \(a^{-n}=\frac{1}{a^n}\) and \(a^{\frac{1}{n}}=\sqrt[n]{a}\). Term 1: \(\frac{1}{2}x^{-1/2}(x+y)^{1/2}=\frac{1}{2}\cdot \frac{1}{x^{1/2}}\cdot (x+y)^{1/2}=\frac{1}{2\sqrt{x}}\sqrt{x+y}\) Term 2: \(\frac{1}{2}x^{1/2}(x+y)^{-1/2}=\frac{1}{2}x^{1/2}\cdot \frac{1}{(x+y)^{1/2}}=\frac{\sqrt{x}}{2\sqrt{x+y}}\)

Step 3: Combine the terms and simplify the expression

Now, let's substitute the simplified terms back into the expression and combined them: \(\frac{\frac{1}{2\sqrt{x}}\sqrt{x+y}-\frac{\sqrt{x}}{2\sqrt{x+y}}}{x + y}=\frac{\sqrt{x+y}-\sqrt{x}}{2\sqrt{x}\sqrt{x+y}}\cdot\frac{1}{x+y}\)

Step 4: Rewrite the simplified expression

Finally, let's rewrite the simplified expression: \(\frac{\sqrt{x+y}-\sqrt{x}}{2\sqrt{x}\sqrt{x+y}(x+y)}\) This is the simplified form of the original expression.

Key Concepts

Simplifying Expressions

When tackling complex expressions, the goal is to make them as simple as possible. For this exercise, the expression contains multiple terms and exponents, which can be intimidating at first glance. However, if we break it down, the process becomes easier. To simplify an expression, follow these steps:

• Identify and separate each term.
• Apply mathematical operations like addition, subtraction, multiplication, or division as required.
• Simplify each term individually by using mathematical rules, such as those for exponents.
• Combine the simplified terms to arrive at a more straightforward expression.

Simplifying doesn't mean altering the meaning or value of the expression; it's about making it easier to read and understand.

Rules of Exponents

Understanding how to manipulate exponents is crucial when simplifying expressions. Exponents are a way to express repeated multiplication of a number by itself. The rules of exponents help you simplify and solve expressions efficiently. Some important rules to remember include:

• If you have a negative exponent, such as \(a^{-n}\), you can rewrite it as \(\frac{1}{a^n}\).
• For fractional exponents, \(a^{\frac{1}{n}}\) translates to the \(n\)-th root of \(a\), or \(\sqrt[n]{a}\).
• When multiplying like bases, add the exponents: \(a^m \cdot a^n = a^{m+n}\).
• When dividing like bases, subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\).

These rules are key to transforming expressions with exponents into simpler forms. Apply them systematically, as shown in the solution steps, to tackle terms separately and bring clarity to the expression.

Mathematical Operations

Performing mathematical operations is at the heart of simplifying expressions. In this exercise, the primary operations are division and the manipulation of terms. Here's a friendly reminder on some principles:

• When you encounter multiple terms within a fraction, it's often helpful to simplify each term before combining them.
• Addition and subtraction involve performing these operations under similar structures or terms, ensuring all terms are accounted for.
• Keep track of operations as you perform them, especially when you're working with exponents, to avoid mistakes.

Using these operations correctly ensures the expression is reduced to its simplest form without changing its original value. The example provided simplifies terms using these operations and exponent rules, demonstrating a clear path from complex to simple.