Problem 2 - Algebra 1: Concepts and Skills

Question

Which of the following is the simplest form of the radical expression \(\frac{4}{\sqrt{3}} ?\) $$A.\quad \frac{4 \sqrt{3}}{9}\quad B.\quad \frac{4 \sqrt{3}}{3} \quad c.\quad \frac{4}{\sqrt{3}}\quad D.\quad \frac{\sqrt{12}}{3}$$

Step-by-Step Solution

Answer

The simplest form of the radical expression \(\frac{4}{\sqrt{3}}\) is \( \frac{4\sqrt{3}}{3} \) (option B).

Step 1: Rationalize the Denominator

Start by multiplying both the numerator and denominator of \(\frac{4}{\sqrt{3}}\) by \(\sqrt{3}\). Doing so will remove the square root from the denominator and is termed as rationalizing the denominator. So, \(\frac{4}{\sqrt{3}}\) becomes \( \frac{4\sqrt{3}}{3} \).

Step 2: Compare the Result with Given Options

Upon reviewing the provided options, it becomes evident that option B, which is \( \frac{4\sqrt{3}}{3} \), matches perfectly with the result derived in Step 1. Therefore, \( \frac{4\sqrt{3}}{3} \) is the simplest form of the given radical expression \(\frac{4}{\sqrt{3}}\).

Key Concepts

Rationalizing the Denominator

Rationalizing the denominator involves a mathematical technique used to eliminate square roots from the denominator of a fraction. This process is crucial because mathematical convention typically prefers fractions without radicals in the denominator for simplification and clarity in expressions.

To rationalize the denominator of a fraction like \(\frac{4}{\sqrt{3}}\), you multiply both the numerator and the denominator by the radical in the denominator to remove it. In this case, you use \(\sqrt{3}\) as a multiplier. This results in:

• Numerator: \(4 \cdot \sqrt{3} = 4\sqrt{3}\)
• Denominator: \(\sqrt{3} \cdot \sqrt{3} = 3\)

Thus, the expression transforms into \(\frac{4\sqrt{3}}{3}\), which now has a rational denominator. This method ensures the expression is in its simplest form, allowing for more straightforward calculation and interpretation.

Square Roots

Square roots are a fundamental concept in mathematics and are vital in the process of simplifying radical expressions. A square root of a number \(x\) is a value that, when multiplied by itself, gives \(x\). The square root of \(x\) is written as \(\sqrt{x}\).

Understanding square roots is essential when working with radicals as they often appear in expressions needing simplification or standardization. For example, when rationalizing a denominator, the square root is multiplied to clear it from the denominator—like in \(\sqrt{3} \cdot \sqrt{3} = 3\).

• This concept demonstrates why multiplying a square root by itself results in a whole number.
• It forms the core principle behind simplification techniques such as rationalizing denominators.

Getting comfortable with square roots ensures a solid foundation in solving quadratic equations and other algebraic expressions contrary to the notion of complex numbers.

Simplifying Radical Expressions

Simplifying radical expressions involves the process of transforming expressions with radicals into their simplest or most understandable form. This includes making sure that no radicals appear in the denominator and that the radical itself is in its most reduced state.

When simplifying radical expressions, consider these steps:

• Combine like terms: For example, \(2\sqrt{a} + 3\sqrt{a} = 5\sqrt{a}\)
• Break down the radicals if possible: \(\sqrt{12}\) can be simplified to \(2\sqrt{3}\) since \(12 = 4 \times 3\), and \(\sqrt{4} = 2\)
• Rationalize the denominator: As shown with \(\frac{4}{\sqrt{3}}\) becoming \(\frac{4\sqrt{3}}{3}\)

Simplifying these expressions is a skill that guarantees working efficiently with more intricate algebraic problems. It not only makes calculations easy but also provides clarity when interpreting mathematical results by reducing them to their most basic forms.