Problem 82 - Algebra and Trigonometry

Question

Simplify the fractional expression. (Expressions like these arise in calculus.) $$ \sqrt{1+\left(x^{3}-\frac{1}{4 x^{3}}\right)^{2}} $$

Step-by-Step Solution

Answer

The simplified expression is \(\sqrt{x^6 + \frac{1}{2} + \frac{1}{16x^6}}\).

Step 1: Identify Components of the Expression

The expression given is \( \sqrt{1+\left(x^{3}-\frac{1}{4 x^{3}}\right)^{2}} \). The key components inside the square root are \(1\) and the square of \(x^{3} - \frac{1}{4 x^{3}}\). Start by noting that \((x^{3} - \frac{1}{4x^{3}})\) is a term we need to simplify through expansion.

Step 2: Expand the Squared Term

Expand the expression \( \left(x^{3}-\frac{1}{4 x^{3}}\right)^{2} \) using the algebraic identity \((a-b)^{2} = a^{2} - 2ab + b^{2}\). Here \(a=x^{3}\) and \(b=\frac{1}{4x^{3}}\). This gives us \(x^6 - 2 \cdot x^3 \cdot \frac{1}{4x^3} + \left(\frac{1}{4x^3}\right)^2\).

Step 3: Simplify the Expanded Terms

Calculating each part: \(x^6\) remains \(x^6\), the middle term simplifies to \(-\frac{2}{4}\) or \(-\frac{1}{2}\), and the last term simplifies to \(\frac{1}{16x^6}\), resulting in \(x^6 - \frac{1}{2} + \frac{1}{16x^6}\).

Step 4: Add and Group Terms Inside the Square Root

The full expression inside the square root becomes \(1 + x^6 - \frac{1}{2} + \frac{1}{16x^6}\). Combining these under a common expression gives \(-\frac{1}{2} + 1 = \frac{1}{2}\), leading to \(\sqrt{x^6 + \frac{1}{2} + \frac{1}{16x^6}}\).

Step 5: Simplify the Expression Further (if Possible)

Since further simplification by factoring or other basic algebraic identities is not applicable without additional context or constraints such as specific values for \(x\), the expression should be noted as simplified to this final form. It is now \(\sqrt{x^6 + \frac{1}{2} + \frac{1}{16x^6}}\).

Key Concepts

Algebraic Identities

Algebraic identities are extremely useful tools that help simplify mathematical expressions. They are pre-established equations that hold true for all values of the variables within them. In this exercise, we apply one of the most straightforward algebraic identities: the square of a difference identity. The identity is written as

• \((a-b)^{2} = a^{2} - 2ab + b^{2}\).

Here, \(a\) and \(b\) can be any numbers or expressions, and using the identity allows us to expand expressions without direct multiplication. This identity is especially handy when dealing with complex terms, such as \( (x^{3}-\frac{1}{4 x^{3}})^{2} \). By substituting \(a = x^3\) and \(b = \frac{1}{4x^3}\), we easily transform the expression into terms that can be more readily simplified. These algebraic shortcuts save time and reduce errors during simplification.

Square Root Simplification

In mathematics, simplifying square roots often reveals simpler forms or insights about the problem. When an expression is under a square root, like

• \(\sqrt{1+(x^{3}-\frac{1}{4 x^{3}})^{2}}\),

our task is to simplify the terms inside first before taking the square root if possible. This approach helps in handling calculations that would become complex if done directly. After expanding the internal expression and summing up similar terms, as demonstrated in the solution, the square root then applies to fewer simplified terms. The key is to ensure all terms inside are in their simplest form before attempting any further operation. Though not all expressions simplify nicely, especially without constraints or specific values for variables, this method still provides a clearer perspective of components involved. Sometimes, like in this exercise, the expression inside the square root remains relatively complex and can be expressed directly afterwards.

Expansion of Expressions

The expansion of expressions, especially when dealing with powers, involves writing them in a multiplied-out form. This is critical in algebra as it allows us to view and rearrange components in simpler terms. For the expression \((x^{3}-\frac{1}{4 x^{3}})^{2}\), expanding it is a crucial step before combining and simplifying further terms. This can be visualized as

• \((x^3)^2 = x^6\)
• \(-2 \cdot x^3 \cdot \frac{1}{4x^3} = -\frac{1}{2}\)
• \((\frac{1}{4x^3})^2 = \frac{1}{16x^6}\).

Upon expanding, each term needs individual attention, where we calculate and reduce them to their simplest state. Only after each part is simplified do we combine like terms. This process not only reduces the complexity of expressions but also preps them for further operations like addition or subtraction, often seen in calculus and other advanced math areas. By mastering expansion, students can more effectively handle a wide range of algebraic problems.