Question

Use a calculator to approximate each radical. Round your answer to two decimal places. $$\frac{2+\sqrt{3}}{3-\sqrt{5}}$$

Short answer

4.89

Step-by-step solution

Simplify the expression using the conjugate

To simplify the given expression, multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of the denominator, \(3 - \sqrt{5}\), is \(3 + \sqrt{5}\). Hence, the expression becomes: \[ \frac{2+\sqrt{3}}{3-\sqrt{5}} * \frac{3+\sqrt{5}}{3+\sqrt{5}} \].

Apply the distributive property (FOIL)

Multiply the numerators and the denominators separately. \[ (2+\sqrt{3})(3+\sqrt{5}) = 2*3 + 2*\sqrt{5} + \sqrt{3}*3 + \sqrt{3}*\sqrt{5} = 6 + 2\sqrt{5} + 3\sqrt{3} + \sqrt{15} \] and \[ (3-\sqrt{5})(3+\sqrt{5}) = 3*3 - (\sqrt{5}*\sqrt{5}) = 9 - 5 = 4 \].

Combine like terms in the numerator

Combine the terms in the numerator obtained from Step 2: \[ 6 + 2\sqrt{5} + 3\sqrt{3} + \sqrt{15} \].

Express the simplified fraction

The simplified expression is now: \[ \frac{6 + 2\sqrt{5} + 3\sqrt{3} + \sqrt{15}}{4} \].

Use a calculator to approximate the radicals

Now, approximate each radical to get a decimal value: \(\sqrt{5} \approx 2.24\), \(\sqrt{3} \approx 1.73\), and \(\sqrt{15} \approx 3.87\).

Substitute the approximate values

Substituting the approximate values into the simplified expression gives: \[ \frac{6 + 2(2.24) + 3(1.73) + 3.87}{4} \].

Simplify the numerator

Calculate the numerator: \[ 6 + 4.48 + 5.19 + 3.87 = 19.54 \].

Divide by the denominator

Finally, divide by 4 to obtain the approximate value: \[ \frac{19.54}{4} = 4.885 \].

Round the result

Round the result to two decimal places: \[ 4.885 \approx 4.89 \].

Key concepts

Rationalizing Denominators

Rationalizing denominators is a technique used in mathematics to eliminate radical expressions, such as square roots, from the bottom (denominator) of a fraction. This process is crucial because it simplifies computations and often makes further mathematical operations more straightforward. In our exercise, we start with an expression where the denominator has a square root, \(3 - \sqrt{5}\).To rationalize this, we multiply both the numerator and the denominator by the 'conjugate' of the denominator. The conjugate’s role is to eradicate the radical. For example, the conjugate of \(3 - \sqrt{5}\) is \(3 + \sqrt{5}\). By multiplying, we use the difference of squares formula, which gets rid of the square root in the denominator. This makes further arithmetic easier because it results in a rational number.

Approximating Radicals

Approximating radicals involves finding a decimal value that is close to the exact value of the square root. This is especially useful when you need a quick or practical solution, and a precise exact answer is not required. In the context of the exercise, we approximate \(\sqrt{3}\), \(\sqrt{5}\), and \(\sqrt{15}\).For instance, \(\sqrt{3}\) can be approximated as 1.73. Although it's not exact, it’s close enough for most practical purposes. We use these approximations to convert the radical expressions into decimal forms, which simplifies calculations significantly. When performing this approximation step with a calculator, it's vital to be aware of the precision needed. Here, we rounded to two decimal places for practicality and simplicity.

Calculator Usage in Mathematics

Calculators are indispensable tools in modern mathematics, aiding in fast and accurate computation of complex expressions. In our exercise, a calculator helps simplify both the radical approximations and the arithmetic operations. Here’s how you can effectively use a calculator for such functions:1. **Square Root Calculations**: Most scientific calculators have a \(\sqrt{ }\) button. To approximate \(\sqrt{5}\), you would enter '5', then press the square root button to get approximately 2.24.2. **Arithmetic Operations**: Calculators also perform basic arithmetic operations (addition, subtraction, multiplication, division). Enter the expression step-by-step to ensure accuracy.By clearly following the calculator steps, you can avoid manual errors and focus on understanding the conceptual aspects of the problem. Using a calculator efficiently can save time and increase the precision of your work.