Problem 54 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Rationalize the denominator of the expression. $$ \frac{3}{\sqrt{5}} $$

Step-by-Step Solution

Answer

The short answer for rationalizing the denominator of the expression \(\frac{3}{\sqrt{5}}\) is to multiply both the numerator and denominator by \(\sqrt{5}\), resulting in the simplified expression \(\frac{3\sqrt{5}}{5}\).

Step 1: Identify the square root in the denominator

In this case, the denominator is \(\sqrt{5}\), and we want to eliminate the square root.

Step 2: Multiply both the numerator and denominator by the square root

To eliminate the square root in the denominator, multiply both the numerator and denominator by \(\sqrt{5}\): \[ \frac{3}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} \]

Step 3: Simplify the expression

Now, multiply the numerators together and the denominators together: \[ \frac{3 \times \sqrt{5}}{\sqrt{5} \times \sqrt{5}} \] This simplifies to: \[ \frac{3\sqrt{5}}{5} \] So, the expression with the rationalized denominator is: \[ \frac{3\sqrt{5}}{5} \]

Key Concepts

Square Roots

Square roots are a fundamental concept in mathematics. They are used to find a value which when multiplied by itself gives the original number. In symbols, the square root of a number \(a\) is written as \(\sqrt{a}\). So, \(\sqrt{25} = 5\) because \(5 \times 5 = 25\). This operation is vital when dealing with irrational numbers, as it helps in simplifying expressions that involve roots.

However, when dealing with fractions, having a square root in the denominator is often not ideal. This is because it can make further calculations cumbersome and unclear. To resolve this, we employ a technique called 'rationalizing the denominator'. This involves eliminating the square root from the denominator of a fraction, making the expression more straightforward to work with.

Multiplication of Fractions

Understanding the multiplication of fractions is key in mathematics. The process is straightforward: you multiply the numerators (the top parts) and then the denominators (the bottom parts) of the fractions together. For example, if you are multiplying \(\frac{3}{4}\) by \(\frac{1}{2}\), you would compute \(3 \times 1\) to get 3, and \(4 \times 2\) to get 8. Thus, the product is \(\frac{3}{8}\).

In the context of rationalizing denominators, this concept applies directly. When you multiply both the numerator and the denominator of a fraction by the same square root, you're essentially multiplying the fraction by 1. This keeps the value of the original fraction unchanged, while allowing you to alter its form. In the exercise, multiplying by \(\frac{\sqrt{5}}{\sqrt{5}}\) enables us to rationalize the expression \(\frac{3}{\sqrt{5}}\) to \(\frac{3\sqrt{5}}{5}\).

Simplifying Expressions

Simplifying expressions makes them easier to work with and understand. This process involves combining like terms and reducing the expression to its simplest form. In cases where fractions are involved, it means reducing ratios to their simplest terms.

In the example given, after rationalizing the denominator by multiplying by the square root, we got \(\frac{3\sqrt{5}}{5}\). This expression cannot be simplified further because the numerator and denominator no longer have common factors. The square root in the numerator is simplified as is, and the fraction is in its lowest terms.

Simplifying expressions is crucial in solving equations, evaluating expressions, and performing operations, as it presents clearer and more manageable results.