Problem 39 - Algebra 2 and Trigonometry

Question

a. Write each fraction in simplest radical form. b. Use a calculator to find a rational approximation for the given fraction. c. Use a calculator to find a rational approximation for the fraction in simplest form. \(\frac{4}{\sqrt{6}}\)

Step-by-Step Solution

Answer

Simplest radical form: \( \frac{2\sqrt{6}}{3} \), approximations: 1.633.

Step 1: Simplify the Radical Fraction

Given the fraction \( \frac{4}{\sqrt{6}} \), we need to simplify it to its simplest radical form. To do this, we rationalize the denominator by multiplying both the numerator and the denominator by \( \sqrt{6} \). This gives us \( \frac{4 \times \sqrt{6}}{\sqrt{6} \times \sqrt{6}} = \frac{4\sqrt{6}}{6} \). We can then simplify this by dividing both the numerator and the denominator by 2: \( \frac{2\sqrt{6}}{3} \). So, the simplest radical form is \( \frac{2\sqrt{6}}{3} \).

Step 2: Rational Approximation of the Original Fraction

Now, use a calculator to find a rational approximation of the original fraction \( \frac{4}{\sqrt{6}} \). First, calculate \( \sqrt{6} \) which is approximately 2.449. Then calculate \( \frac{4}{2.449} \), which is approximately 1.633. Thus, the rational approximation of the original fraction is 1.633.

Step 3: Rational Approximation of the Simplified Form

Next, use a calculator to approximate the simplified radical form \( \frac{2\sqrt{6}}{3} \). Calculate \( 2 \times \sqrt{6} \) which is approximately 4.899. Then calculate \( \frac{4.899}{3} \), which is approximately 1.633. Therefore, the rational approximation for the fraction in simplest form is also 1.633.

Key Concepts

Simplifying Fractions

Simplifying fractions is an essential concept that helps in making mathematical problems easier to deal with. When it comes to fractions involving radicals, the goal is to express the fraction in its simplest form by reducing the values in both the numerator and the denominator.
To simplify a fraction with a radical, like \( \frac{4}{\sqrt{6}} \), we look for opportunities to clear or reduce the radical in the denominator. This often involves rationalizing the denominator, which we'll discuss more below. Once the denominator is free of radicals, check to simplify the fraction further by dividing both the numerator and the denominator by their greatest common factor.
For example, after rationalizing the denominator of \( \frac{4}{\sqrt{6}} \), we obtain \( \frac{4\sqrt{6}}{6} \). Dividing the numerator and denominator by 2 further simplifies this to \( \frac{2\sqrt{6}}{3} \). These steps help ensure that the fraction is presented in its simplest radical form, which is easier to evaluate and use in further calculations.

Rationalizing the Denominator

Rationalizing the denominator is crucial when dealing with radical fractions. The aim is to eliminate any radical expressions in the denominator to create a simpler and more usable fraction.
A denominator is rationalized by multiplying both the numerator and the denominator by the radical found in the denominator. For example, in the fraction \( \frac{4}{\sqrt{6}} \), we multiply both the numerator and the denominator by \( \sqrt{6} \). This results in \( \frac{4\sqrt{6}}{6} \).

• Why Rationalize? Removing the radical from the denominator simplifies arithmetic operations, especially when adding or subtracting fractions.
• General Method: Multiply by a form of 1, here it's \( \frac{\sqrt{6}}{\sqrt{6}} \). It adjusts the fraction without changing its value.

Rationalization not only simplifies calculations but also makes the fraction more conventional, aligning with the mathematical standard to avoid radicals in denominators.

Rational Approximation

Rational approximation involves finding a decimal number that is close to the exact value of a radical fraction. This method is especially helpful when precise calculations are required, or when using the fraction in practical real-world applications.
To find a rational approximation, we use a calculator to solve and convert radicals and fractions to decimal form. For the original fraction \( \frac{4}{\sqrt{6}} \), calculate \( \sqrt{6} \approx 2.449 \), and then \( \frac{4}{2.449} \approx 1.633 \). This decimal, 1.633, represents a close approximation of the original radical fraction.
When the fraction has been simplified, such as \( \frac{2\sqrt{6}}{3} \), calculate it by multiplying \( 2 \times \sqrt{6} \approx 4.899 \), then \( \frac{4.899}{3} \approx 1.633 \). Interestingly, the approximation remains the same, showing consistency across forms.

• Why Use Rational Approximation? Simplifies further calculations, especially when precision isn't paramount but usability is.
• Practical Uses: Useful in engineering, physics, and applied mathematics where exactness might be less critical than usability.

Thus, rational approximations are an everyday tool to swiftly convert complex-looking fractions into more understandable figures, streamlining mathematical interpretation and application.