Question

Simplify the expression. $$2 \sqrt{\frac{5}{4}}$$

Short answer

\(\sqrt{5}\)

Step-by-step solution

Apply the Property of Radicals

We first apply the property of radicals, which states that the square root of a quotient is the quotient of the square roots. This gives us: \(2 \cdot \sqrt{\frac{5}{4}} = 2 \cdot \frac{\sqrt{5}}{\sqrt{4}}\).

Simplify the Square Root

The square root of 4 is 2. Therefore, we simplify \(\sqrt{4}\) to 2, which gives us: \(2 \cdot \frac{\sqrt{5}}{2}\).

Cancel Common Factors

The 2 in the numerator and the 2 in the denominator cancel out, leaving us with: \(\sqrt{5}\).

Key concepts

Property of Radicals

When simplifying radical expressions, it's essential to understand the property of radicals. This property is fundamental in algebra and significantly simplifies complex-looking equations into more manageable forms. Here's the concept: the square root of a quotient (a division) is equal to the quotient of the square roots. In simpler terms, when you have a fraction under a square root sign, you can take the square root of the numerator and denominator separately.

For example, when simplifying \( \sqrt{\frac{a}{b}} \), you can change this to \( \frac{\sqrt{a}}{\sqrt{b}} \). Utilizing this property lets you break down seemingly difficult expressions into smaller parts that are easier to deal with, as you can individually simplify the square roots of the numerator and the denominator.

Square Root Simplification

To square root simplification, it's all about converting a radical to its simplest form. Often, you'll find that square roots can actually be simplified by recognizing perfect squares. A perfect square is a number like 4, 9, 16, and so on, where the square root results in a whole number. So, if you see \( \sqrt{4} \), you immediately know that it can be simplified to 2 because 2 squared (2 times 2) equals 4.

By identifying and extracting perfect squares from under the radical, you not only make the expression simpler but also often turn irrational numbers (numbers that can't be precisely expressed as a simple fraction) into rational numbers, which are easier to work with in calculations.

Simplifying Square Roots of Quotients

When you come across simplifying square roots of quotients, it's just a combination of the previous concepts. You first apply the property of radicals to the quotient, which allows you to handle the numerator and the denominator separately. After breaking them down, you can look at each part and simplify any square roots of perfect squares in the individual numerator and denominator.

Let's take an example. Starting with \(2 \sqrt{\frac{5}{4}}\), you would first convert it into \(2 \cdot \frac{\sqrt{5}}{\sqrt{4}}\) and then simplify \(\sqrt{4}\) to 2 because 4 is a perfect square. Lastly, to simplify the expression, you would cancel out any common factors - just like the 2's in the numerator and denominator here. The end result is a much cleaner, simplified form that's much easier to understand and utilize in further calculations or applications.