Problem 108 - Elementary and Intermediate Algebra

Question

The following problems involve addition, subtraction, and multiplication of radical expressions, as well as rationalizing the denominator. Perform the operations and simplify, if possible. All variables represent positive real numbers. $$ \frac{\sqrt{3}}{\sqrt{98 x^{2}}} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{\sqrt{6}}{14x}\).

Step 1: Simplify the Radical in the Denominator

We begin by simplifying the expression under the square root in the denominator. Notice that \(98x^2\) can be broken down into prime factors as \(98 = 2 \times 7^2\). Thus, we write: \(\sqrt{98x^2} = \sqrt{2 \times 7^2 \times x^2}\).

Step 2: Separate the Radical Terms

Now, apply the property \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\): \(\sqrt{98x^2} = \sqrt{2} \cdot \sqrt{7^2} \cdot \sqrt{x^2}\).

Step 3: Simplify Each Separate Term

Simplify each radical term: \(\sqrt{7^2} = 7\) and \(\sqrt{x^2} = x\). Thus, \(\sqrt{98x^2} = \sqrt{2} \times 7 \times x = 7x\sqrt{2}\).

Step 4: Rationalize the Denominator

To rationalize the denominator, we need to eliminate \(\sqrt{2}\). Multiply both the numerator and the denominator by \(\sqrt{2}\): \(\frac{\sqrt{3}}{7x\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{6}}{14x} .\)

Step 5: Writing the Final Simplified Expression

The expression \(\frac{\sqrt{6}}{14x}\) is now fully simplified, with no radicals in the denominator. This concludes our work on simplifying the given radical expression.

Key Concepts

Simplifying Radicals

Simplifying radicals is an essential skill when working with radical expressions, which frequently appear in various mathematical contexts. The process involves breaking down the expression into its simplest form.

To simplify a radical, you need to look for perfect squares (for square roots) or other powers according to the type of root you're dealing with inside the radical. This means finding parts of the radicand that can be removed from under the radical sign.

For example, in the expression provided, \(98x^2\) is factorized into \(98 = 2 \times 7^2\) and \(x^2\). Using properties of radicals, specifically the property \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\), we separate each factor: \(\sqrt{2} \cdot \sqrt{7^2} \cdot \sqrt{x^2}\).

• \(\sqrt{7^2} = 7\) because \(7^2\) is a perfect square.
• \(\sqrt{x^2} = x\) as \(x^2\) simplifies to \(x\).

The radical simplifies to \(7x\sqrt{2}\), effectively reducing the complexity of the original expression. By simplifying, calculations become easier and clearer in both the numerator and denominator.

Rationalizing the Denominator

Rationalizing the denominator is the process of eliminating any radicals present in the denominator of a fraction. This is a common requirement for simplifying expressions, as a denominator with radicals can make it less straightforward to understand or use computationally.

To rationalize a denominator like \(7x\sqrt{2}\), identified in the step-by-step solution, multiply both the numerator and the denominator by \(\sqrt{2}\).

Here's the reasoning:

• Multiply: \(\frac{\sqrt{3}}{7x\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{3}\sqrt{2}}{7x \sqrt{2}\sqrt{2}}\).
• Apply the property that \(\sqrt{a} \cdot \sqrt{a} = a\). Hence, \(\sqrt{2} \cdot \sqrt{2} = 2\).

This multiplication results in \(\frac{\sqrt{6}}{14x}\). The radicals in the denominator are thus removed, achieving a more standard fractional form. For simplification purposes and ease of further computations, it's always a good practice to rationalize denominators. It transforms fractions into a more manageable form and aligns with mathematical standards.

Algebraic Expressions

Algebraic expressions form the backbone of much of algebra and higher mathematics. They consist of variables, numbers, and operations. In the context of the problem solved, algebraic expressions come into play when dealing with both the radical components and the expressions simplified into fractions.

Understanding the structure and manipulation of these expressions is crucial. Here are a few key points:

• Variables, such as \(x\), represent unknown values and can be manipulated just like numbers.
• Operations can include addition, subtraction, multiplication, division, and exponentiation.
• The expressions can be simplified by grouping like terms and applying algebraic rules, such as distributive and associative properties.

In the given solution, the expression \(\frac{\sqrt{6}}{14x}\) features both a numerical part and an algebraic part (involving \(x\)). Recognizing the role of algebraic expressions helps in applying operations like rationalization correctly and understanding the resulting expressions' implications. Each manipulation adheres to the rules governing algebraic structures, ensuring that expressions remain valid across transformations.