Problem 69 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Write the expression in simplest radical form. $$ \sqrt{\frac{16}{3}} $$

Step-by-Step Solution

Answer

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

Step 1: Find the square root of the numerator

Find the square root of 16. \( \sqrt{16}=\)

Step 2: Find the square root of the denominator

Find the square root of 3. \( \sqrt{3}=\)

Step 3: Write the simplified expression

Combine the results of Step 1 and Step 2 to write the expression in simplest radical form. \( \sqrt{\frac{16}{3}}=\frac{\sqrt{16}}{\sqrt{3}} \)

Step 4: Simplify the expression

Simplify the expression by substituting the results from Step 1 and Step 2. \( \frac{4}{\sqrt{3}} \)

Step 5: Rationalize the denominator

To rationalize the denominator, multiply the numerator and denominator by the conjugate of the denominator, which is \(\sqrt{3}\) in this case. \( \frac{4}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{4\sqrt{3}}{3} \) The simplest radical form of the given expression is: \( \boxed{\frac{4\sqrt{3}}{3}} \)

Key Concepts

Square Roots

Square roots are mathematical expressions used to represent a value that, when multiplied by itself, equals the original number. For instance, the square root of 16, written as \( \sqrt{16} \), equals 4 since \( 4 \times 4 = 16 \). Square roots play a critical role in simplifying radical expressions as they help to break down complex numbers into simpler forms.

When dealing with fractions inside a square root, such as \( \sqrt{\frac{16}{3}} \), we can simplify them by finding the square roots of both the numerator (16) and the denominator (3) separately. Breaking it down this way allows us to manage each part of the fraction more easily.

Here's how you perform it step by step:

• Calculate \( \sqrt{16} \) which equals 4.
• The \( \sqrt{3} \) remains as it is because 3 is not a perfect square.

This gives us part of the expression \( \frac{\sqrt{16}}{\sqrt{3}} \), which is simplified to \( \frac{4}{\sqrt{3}} \). Understanding square roots is essential to succeed in simplifying such radical expressions.

Rationalizing the Denominator

Rationalizing the denominator refers to the process of eliminating any irrational numbers, such as square roots, from the denominator of a fraction. Since fractions are easier to work with when they don't have square roots in the denominator, this step is often crucial in the simplification process.

In our example, we start with \( \frac{4}{\sqrt{3}} \). The denominator here, \( \sqrt{3} \), is irrational. Here’s how we can rationalize it:

• Multiply both the numerator and the denominator by \( \sqrt{3} \) to make the denominator rational.

By performing \( \frac{4}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} \), we get:
\( \frac{4\sqrt{3}}{3} \).

Through this method, the denominator becomes 3, which is a rational number. This makes the expression easier to work with while preserving its original value. Rationalizing the denominator is a valuable skill as it simplifies and standardizes the way fractions are presented.

Expression Simplification

Simplifying expressions involves making them easier to read and work with, without altering their value. This process often includes breaking complex expressions into smaller, more manageable pieces, and then performing operations such as rationalizing.

In the expression \( \sqrt{\frac{16}{3}} \), we followed a systematic approach to simplify it:

• First, calculate the square root of the numerator and the denominator, yielding \( \frac{\sqrt{16}}{\sqrt{3}} = \frac{4}{\sqrt{3}} \).
• Next, rationalize the denominator by multiplying by \( \frac{\sqrt{3}}{\sqrt{3}} \) to get \( \frac{4\sqrt{3}}{3} \).

This practice of simplification is foundational in algebra. It not only spikes the accuracy of computations but also enhances the clarity of mathematical communication. By following these steps, you can transform an expression into its simplest radical form, ensuring it's both standardized and efficient for further mathematical operations.