Problem 88 - Algebra and Trigonometry

Question

Factor the expression completely. Begin by factoring out the lowest power of each common factor. $$ x^{-1 / 2}(x+1)^{1 / 2}+x^{1 / 2}(x+1)^{-1 / 2} $$

Step-by-Step Solution

Answer

The factored expression is \(x^{-1/2}(x+1)^{-1/2}(2x + 1)\).

Step 1: Identify the Common Factor

Examine each term in the expression \(x^{-1 / 2}(x+1)^{1 / 2}+x^{1 / 2}(x+1)^{-1 / 2}\). Notice that both terms have common factors involving powers of \(x\) and \((x+1)\). Specifically, \(x^{-1/2}\) is the lowest power of \(x\) across both terms, and \((x+1)^{-1/2}\) is the lowest power of \((x+1)\).

Step 2: Factor Out the Lowest Power of Each Common Factor

Factor \(x^{-1/2}\) and \((x+1)^{-1/2}\) from both terms. The expression becomes:\[x^{-1/2}(x+1)^{-1/2}\left[(x+1) + x\right].\]

Step 3: Simplify the Expression Inside the Brackets

Simplify the expression inside the brackets, \((x+1) + x = 2x + 1\).

Step 4: Write the Factored Expression

Combine all pieces to rewrite the entire expression as:\[x^{-1/2}(x+1)^{-1/2}(2x + 1).\]

Key Concepts

Understanding Common Factor

When dealing with algebraic expressions, identifying a common factor is essential for simplifying them. A common factor is a term that divides each part of an expression evenly. In our exercise, we are given

• \( x^{-1 / 2}(x+1)^{1 / 2} + x^{1 / 2}(x+1)^{-1 / 2} \)

Here, we notice that both terms share certain components:

• The variable \( x \)
• The expression \((x+1)\)

Now, the key is to find the lowest power of these components as a common factor. In this case, \( x^{-1 / 2} \) and \((x+1)^{-1 / 2}\) are extracted because they appear in each term with the smallest exponents. By factoring these out, you can more clearly see the structure of the expression before further simplification.

Exploring Powers of x

Powers of x, or more generally, exponents in algebra, tell us how many times a number (the base) is multiplied by itself. They can complicate an expression, but also provide a useful tool for simplification. In our expression,

• \( x^{-1 / 2} \) represents the square root of \( x \) in the denominator.
• \( x^{1 / 2} \) is the square root of \( x \) in the numerator.

Recognizing these powers allows us to manipulate the expression strategically. When factoring, use the lowest possible power. In this problem, choosing the lowest exponent \( x^{-1/2} \) simplifies factoring, as it can be factored out from both terms and simplify the multiplication inside the expression.

Simplifying Expressions Efficiently

Simplifying expressions is a critical step in making them more manageable and understable. After factoring out the common factor in our given expression, we're left with

• \[ x^{-1/2}(x+1)^{-1/2}((x+1) + x) \]

To simplify, you need to combine like terms inside the brackets. Here, this means adding \(x+1\) and \(x\), which simplifies to \(2x+1\). This gives us the expression

• \[ x^{-1/2}(x+1)^{-1/2}(2x+1). \]

Notice how the expression is now much clearer and simpler. Here, careful factoring and manipulating powers helped streamline a potentially complicated expression, making further calculations or interpretations easier.