Problem 93 - College Algebra

Question

In Exercises 85-94, factor and simplify each algebraic expression. $$(4 x-1)^{1 / 2}-\frac{1}{3}(4 x-1)^{3 / 2}$$

Step-by-Step Solution

Answer

The factored and simplified form of the given algebraic expression is \[\frac{4(4x - 1)^{1 / 2} - 4x(4x - 1)^{1 / 2}}{3}\]

Step 1: Identify Common Base

Look for a common base in the given terms. Here, both terms contain \((4x - 1)\) with different exponents.

Step 2: Use the Distributive Property

Let's take the part of the terms with the lowest exponent \((4x - 1)^{1 / 2}\) out of the two terms to use the distributive property. It would look like this: \[(4 x-1)^{1 / 2}(1 - \frac{1}{3}(4 x-1)) \]

Step 3: Simplify the Extracted Polynomial

Now, simplify the extracted polynomial within the parentheses. This will give: \[(4 x-1)^{1 / 2}(1 - \frac{4x}{3} + \frac{1}{3})\]. Going further this simplifies to: \[(4x -1)^{1 / 2}\*(\frac{-4x + 4}{3})\]

Step 4: Distribute the Term Outside the Parentheses

Now, let's distribute the term outside the parentheses: \[\frac{-4x(4x - 1)^{1 / 2} + 4(4x -1)^{1 / 2}}{3}\]

Step 5: Rearrange the Expression

The expression can be rearranged to make the terms look simpler: \[\frac{4(4x - 1)^{1 / 2} - 4x(4x - 1)^{1 / 2}}{3}\]

Key Concepts

Understanding the Distributive Property

The distributive property is a key concept in algebra that helps us simplify expressions by distributing a factor across terms inside parentheses. In this exercise, we identified a common factor, \((4x - 1)^{1/2}\), and used it to make the expression easier to handle.

Imagine you have an expression like \(a(b + c)\). The distributive property allows you to write this as \(ab + ac\).

In our problem, we took \((4x - 1)^{1/2}\) out of the two terms, making the whole expression more manageable:

• Original: \((4x - 1)^{1/2} - \frac{1}{3}(4x - 1)^{3/2}\)
• Factored: \((4x - 1)^{1/2}(1 - \frac{1}{3}(4x - 1))\)

This simplification is crucial for making complex expressions less intimidating.

The Art of Factoring in Algebra

Factoring is about breaking down expressions into multipliers or simpler elements. In algebra, recognizing common factors can significantly simplify expressions.

In this exercise, we spotted a common factor of \((4x - 1)^{1/2}\) in both terms. By factoring it out, the expression becomes:

• From \((4x - 1)^{1/2} - \frac{1}{3}(4x - 1)^{3/2}\)
• To \((4x - 1)^{1/2}(1 - \frac{4x}{3} + \frac{1}{3})\)

By pulling out the common factor, we transformed a complex expression into one that's easier to work with.

Factoring allows us to see the structure more clearly, making further simplifications smoother.

Navigating Exponents With Confidence

Exponents indicate how many times a number or expression is multiplied by itself. They are essential in algebraic expressions, and understanding them is crucial for solving problems efficiently.

In our exercise, we dealt with terms like \((4x - 1)^{1/2}\) and \((4x - 1)^{3/2}\). The exponents tell us the power to which the base, \(4x - 1\), is raised:

• \((4x - 1)^{1/2}\) means the square root of \(4x - 1\).
• \((4x - 1)^{3/2}\) can be seen as \((4x - 1)\times\sqrt{4x - 1}\).

Handling different exponents correctly allowed us to factor and organize the terms, revealing a pathway to a simpler solution.

Simplification: The Final Touch

Simplification is the process of making an expression easier to read and work with. After factoring and applying properties, simplification is the final step to ensure that the expression is in its simplest form.

We started with:
\[(4x - 1)^{1/2}(1 - \frac{4x}{3} + \frac{1}{3})\]

And simplified it to:

• \[(4x - 1)^{1/2}\times\frac{-4x + 4}{3}\]

Finally, we distributed and rearranged if needed:
\[\frac{4(4x - 1)^{1/2} - 4x(4x - 1)^{1/2}}{3}\]

Simplification is key as it ensures the expression is concise and ready for any further mathematical operations or interpretations.