Problem 39 - Intermediate Algebra

Question

Add or subtract. $$ \sqrt{\frac{28}{x^{2}}}+\sqrt{\frac{7}{4 x^{2}}} $$

Step-by-Step Solution

Answer

\( \frac{5\sqrt{7}}{2x} \)

Step 1: Simplify Each Radical

Begin by simplifying each radical separately. The first radical is \( \sqrt{\frac{28}{x^2}} \), which can be rewritten as \( \frac{\sqrt{28}}{x} \) because \( \sqrt{x^2} = x \). The second radical \( \sqrt{\frac{7}{4x^2}} \) can be rewritten as \( \frac{\sqrt{7}}{2x} \) since \( \sqrt{4x^2} = 2x \).

Step 2: Simplify Radicals Further

Further simplify the radicals: \( \sqrt{28} = \sqrt{4 \times 7} = 2\sqrt{7} \) and \( \sqrt{7} \) stays as is. Thus, we have \( \frac{2\sqrt{7}}{x} + \frac{\sqrt{7}}{2x} \).

Step 3: Combine Fractions

Find a common denominator to add the fractions. The least common denominator for \( x \) and \( 2x \) is \( 2x \). Rewrite each fraction: \( \frac{4\sqrt{7}}{2x} \) (by multiplying \( \frac{2\sqrt{7}}{x} \) by \( 2/2 \)) and \( \frac{\sqrt{7}}{2x} \) remains the same.

Step 4: Add the Fractions

Now, add the fractions: \( \frac{4\sqrt{7}}{2x} + \frac{\sqrt{7}}{2x} = \frac{(4\sqrt{7} + \sqrt{7})}{2x} \). Factor out \( \sqrt{7} \) from the numerator: \( \frac{\sqrt{7}(4 + 1)}{2x} \), which simplifies to \( \frac{5\sqrt{7}}{2x} \).

Key Concepts

Understanding Algebraic Fractions

Algebraic fractions are fractions where the numerator, denominator, or both contain algebraic expressions. In the context of the problem, both fractions are algebraic because they involve variables, specifically "x". Simplifying algebraic fractions involves:

• Breaking down complex expressions into simpler components.
• Applying basic algebraic identities and properties.

Here, the process begins by rewriting the radicals so the components are clearer, transforming them into more manageable forms, such as expressing square roots of variables. This helps in applying further operations like addition or subtraction.

Breaking Down Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, etc. Simplifying them can be daunting at first. In this exercise, we focus on square roots. To simplify the expression \( \sqrt{\frac{28}{x^2}} \) to \( \frac{\sqrt{28}}{x} \), we separate the square root of the fraction into a division of square roots.

To make it simpler:

• Focus on making the denominator a simple expression, \( x \), by taking the square root of \( x^2 \).
• For \( \sqrt{28} \), break it down to its prime factors. Since \( 28 = 4 \times 7 \), and \( \sqrt{4} = 2 \), we can simplify it further to \( 2\sqrt{7} \).

This step is crucial as it lays the foundation for combining the radical expressions later by finding a common denominator.

Finding a Common Denominator

Combining fractions requires a common denominator. For the radicals \( \frac{2\sqrt{7}}{x} \) and \( \frac{\sqrt{7}}{2x} \), the least common denominator (LCD) is \( 2x \). Achieving a common denominator involves:

• Identifying the least common multiple of the denominators.
• Adjusting each fraction accordingly to share this common measure.

Here, you multiply \( \frac{2\sqrt{7}}{x} \) by \( \frac{2}{2} \) to make the denominator \( 2x \), aligning it with \( \frac{\sqrt{7}}{2x} \). With a shared denominator, you set the stage for direct combination of the numerators while keeping the denominator consistent.

The Simplicity of Fraction Addition

Once both fractions \( \frac{4\sqrt{7}}{2x} \) and \( \frac{\sqrt{7}}{2x} \) have a common denominator, they can be easily added together. In essence, adding fractions with the same denominator is straightforward:

• Keep the denominator as it is.
• Add the numerators.

Thus, the sum is \( \frac{(4\sqrt{7} + \sqrt{7})}{2x} \). Notice how we factor out \( \sqrt{7} \) from the numerator, simplifying it to \( \frac{5\sqrt{7}}{2x} \). This simplification not only makes the final expression neater but also highlights the underlying patterns in algebraic manipulation.