Question

Rationalize the numerator of each expression. Assume that all variables are positive when they appear. $$\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}}$$

Short answer

1 + \(\frac{\sqrt{15}}{5}\)

Step-by-step solution

- Identify the given expression

The given expression is \(\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5}}\).

- Multiply the numerator and the denominator by the conjugate of the numerator

To rationalize the numerator, multiply both the numerator and the denominator by \(\sqrt{5}\). This will eliminate the square root in the numerator.

- Simplify the multiplication

Multiply the numerator and the denominator separately: Numerator: \((\sqrt{5} + \sqrt{3}) \cdot \sqrt{5} = \sqrt{5} \cdot \sqrt{5} + \sqrt{3} \cdot \sqrt{5} = 5 + \sqrt{15}\)Denominator: \(\sqrt{5} \cdot \sqrt{5} = 5\)

- Write the final expression

The expression now is \(\frac{5 + \sqrt{15}}{5}\).

- Simplify the expression

Since both terms in the numerator are divisible by 5, simplify the expression to \(1 + \frac{\sqrt{15}}{5}\).

Key concepts

Rationalization

Rationalization is a key technique used to simplify expressions involving square roots. When you rationalize a fraction, you essentially eliminate the square roots from the denominator or numerator. This makes the expression easier to handle.

In the given exercise, the goal was to rationalize the numerator. By multiplying both the numerator and the denominator by the same value, we were able to remove the square root from the numerator effectively. Rationalization often involves the use of conjugates, especially when dealing with expressions that have two terms involving square roots.

Numerator

The numerator is the top part of a fraction. In this exercise, the initial numerator was \(\sqrt{5} + \sqrt{3}\). Rationalizing means we wanted to eliminate the square roots from this part of the equation.

To do this, we multiplied both the numerator and the denominator by \(\sqrt{5}\). This choice is strategic. By choosing \(\sqrt{5}\), we align with the terms present in the numerator, simplifying the whole process.

After multiplication, our new numerator became \(\sqrt{5} \times \sqrt{5} + \sqrt{3} \times \sqrt{5} = 5 + \sqrt{15}\). This form removes individual square roots, aiding further simplification.

Square Roots

Square roots are numbers that produce a certain value when multiplied by themselves. They often appear in algebraic expressions and can make simplification challenging.

In rationalizing the numerator, we tackled square roots directly. The original numerator had \(\sqrt{5} + \sqrt{3}\). By multiplying by \(\sqrt{5}\), we used the property that the product of two identical square roots equals the number itself (\( \sqrt{a} \times \sqrt{a} = a \)). This property helped us turn \(\sqrt{5} \times \sqrt{5} = 5\).

Managing these square roots skillfully reduces complexity in algebraic expressions.

Algebraic Simplification

Algebraic simplification involves breaking down complex expressions into simpler forms. Through rationalization in this exercise, we aimed for a simplified numerator.

After multiplying and obtaining \(\frac{5 + \sqrt{15}}{5}\), simplification revealed the final form as \(\frac{5}{5} + \frac{\sqrt{15}}{5} = 1 + \frac{\sqrt{15}}{5}\). Each term in the numerator was divisible by the denominator. Simplifying fractions this way ensures that we achieve the most reduced form of the expression.

Following these steps consistently in algebra helps in handling a variety of equations more efficiently.