Problem 25 - Algebra for College Students

Question

Express each of the following in simplest radical form. All variables represent positive real numbers. \(\sqrt{20 x^{2} y}\)

Step-by-Step Solution

Answer

\(2x\sqrt{5y}\)

Step 1: Identify the components inside the square root

We begin by identifying what's inside the square root. We have the expression \(\sqrt{20x^2y}\). This can be broken down into the product: \(20 \cdot x^2 \cdot y\).

Step 2: Simplify the square root of each component

Next, we simplify each part under the square root separately. For \(20\), which is \(4 \times 5\), we know \(\sqrt{4} = 2\). For \(x^2\), we have \(\sqrt{x^2} = x\) since \(x\) is a positive real number.

Step 3: Combine simplified forms

Combine the simplified forms: \(\sqrt{20x^2y} = \sqrt{4 \times 5 \times x^2 \times y} = \sqrt{4} \times \sqrt{x^2} \times \sqrt{5y}\). This gives us \(2x\sqrt{5y}\) when simplified.

Key Concepts

Square Root Simplification

Simplifying square roots involves breaking down the expression inside the square root sign into its simplest form. In the example, we worked with \(\sqrt{20x^2y}\).
First, we look at each component—numbers and variables—individually. For numbers, our goal is to find perfect squares, if possible. Here, \(20\) can be split into \(4 \times 5\), where \(4\) is a perfect square.
After that, we take variables. For \(x^2\), the square root is \(x\), as we know \(x\) is positive. For our mixed number and variable \(5y\), there are no perfect squares, so it stays under the square root.
The simplified result is \(2x\sqrt{5y}\). By handling numbers and variables separately, we ensure our expression is in its simplest radical form. Remember, the key is identifying perfect squares.

Positive Real Numbers

In mathematics, we often deal with real numbers, which include both positive and negative numbers as well as zero. However, when working with radicals, especially square roots, it’s important to focus on positive real numbers.
Positive real numbers are greater than zero and are crucial in radical expressions to avoid dealing with imaginary numbers. In exercises like our example \(\sqrt{20x^2y}\), the assumption is made that variables represent positive numbers; this simplifies the square root process.

• If \(x\) is positive, \(\sqrt{x^2}\) becomes \(x\).
• This assumption ensures that the square root calculations are straightforward, avoiding issues like undefined results.

In simple terms, emphasizing positivity ensures clarity and correctness in math problems involving radicals.

Radical Expressions

Radical expressions involve roots, with the square root being the most common. The term 'radical' comes from the Latin 'radix,' meaning root. When you see the square root symbol \(\sqrt{}\), it's technically referred to as a radical.
Simplifying radical expressions is about reducing the terms inside to their simplest forms, much like our example \(\sqrt{20x^2y}\). It's all about recognizing factors and pulling out perfect squares.

• For numbers, seek perfect square factors.
• For variables, identify and simplify powers as appropriate.

The purpose is to express radicals in a way that's easy to understand and use. In mathematics, when expressions are simplified, they become cleaner and more manageable. This principle is vital across all areas, not just square roots, making problem-solving more intuitive.