Problem 96 - Algebra and Trigonometry

Question

Factor the expression completely. (This type of expression arises in calculus in using the “product rule.”) $$ \frac{1}{2} X^{-1 / 2}(3 X+4)^{1 / 2}+\frac{3}{2} X^{1 / 2}(3 X+4)^{-1 / 2} $$

Step-by-Step Solution

Answer

Factored expression: \(X^{-1/2}(3X+4)^{-1/2}(3X + 2)\).

Step 1: Identify the common factor

Both terms in the expression have similar factors. The first term contains \(X^{-1/2}\) and the second term \(X^{1/2}\), allowing us to factor out \(X^{-1/2}\). Both terms also involve \((3X + 4)\); with exponents \((3X + 4)^{1/2}\) and \((3X + 4)^{-1/2}\), we can factor out \((3X+4)^{-1/2}\) from both terms.

Step 2: Factor the expression

Factor out the common factor \(X^{-1/2}(3X+4)^{-1/2}\) from the expression. This gives:\[X^{-1/2}(3X+4)^{-1/2} \left( \frac{1}{2}(3X+4) + \frac{3}{2}X \right)\]

Step 3: Simplify the expression inside the parentheses

Distribute within the parentheses:\[\frac{1}{2}(3X+4) + \frac{3}{2}X = \frac{1}{2} \cdot 3X + \frac{1}{2} \cdot 4 + \frac{3}{2}X\]Simplifying further:\[\frac{3}{2}X + 2 + \frac{3}{2}X = 3X + 2\]

Step 4: Write the final factored expression

Insert the simplified expression from Step 3 back into the expression from Step 2. The completely factored expression is:\[X^{-1/2}(3X+4)^{-1/2}(3X + 2)\]

Key Concepts

Common Factor

Factoring out the common factor is a critical step when simplifying algebraic expressions. Think of it like finding a common thread that runs through each term of an expression. A common factor is anything that appears in every term, making it possible to "take it outside" the expression.

To identify the common factor, look for common numerical or algebraic elements, such as coefficients or variables raised to an exponent. In our example, both terms contain powers of \(X\) and \((3X + 4)\). Specifically, they share the factors \(X^{-1/2}\) and \((3X + 4)^{-1/2}\). By factoring these out from each term, you simplify the expression significantly.

Why is this useful? By simplifying, you make it easier to work with the expression, potentially saving time and avoiding mistakes in complex calculations. Always check for the greatest factor first, as factoring out common parts is one of the first steps in any algebraic simplification.

Exponent Properties

Understanding exponent properties is essential for manipulating expressions with power-based terms. An exponent tells you how many times a number, known as the base, is multiplied by itself. There are several key rules when working with exponents:

• Product of Powers Rule: When multiplying like bases, you add their exponents: \(a^m \times a^n = a^{m+n}\).
• Power of a Power Rule: When taking a power to another power, multiply the exponents: \((a^m)^n = a^{m\cdot n}\).
• Negative Exponent Rule: A negative exponent indicates a reciprocal: \(a^{-n} = \frac{1}{a^n}\).

In the given exercise, these rules help simplify the powers involved, like going from \(X^{-1/2}\) and \((3X+4)^{-1/2}\) to their factored form. By becoming familiar with these properties, you can deftly navigate through tasks involving more complicated expressions, ensuring that errors from exponent manipulation are minimized.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves a combination of identifying common factors and applying properties of exponents, among other techniques. The goal is to rewrite the expression in its simplest form.

Begin by combining like terms and reducing any complex fractions or expressions. In our example, after factoring out the common factor, we simplify what's left inside the parentheses: \(\frac{1}{2}(3X+4) + \frac{3}{2}X\). This means distributing and combining terms to simplify further, reaching \(3X + 2\).

Simplification not only makes the expression easier to handle but also prepares it for further algebraic procedures, such as solving, graphing, or evaluating, where a simpler form is often more beneficial. Remember, the more you practice, the more patterns you'll recognize, enabling you to simplify expressions with ease and accuracy.