Problem 15 - Prealgebra

Question

Simplify each expression by taking as much out from under the radical as possible. You may assume that all variables represent positive numbers $$\sqrt{50 x^{2}}$$

Step-by-Step Solution

Answer

The simplified expression is \(5x\sqrt{2}\).

Step 1: Identify Perfect Squares

The number 50 can be factored into 25 and 2, where 25 is a perfect square. Thus, we write 50 as \( 25 \times 2 \).

Step 2: Simplify the Radical

Using the property \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \), we can write \( \sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2} \).

Step 3: Simplify the Variable Part

Apply the square root property \( \sqrt{x^2} = x \). So, \( \sqrt{x^2} = x \).

Step 4: Combine Parts

Combine the simplified numerical and variable parts: \( 5\sqrt{2} \times x = 5x\sqrt{2} \).

Key Concepts

Perfect Squares

Perfect squares are numbers or expressions that can be expressed as the square of an integer. In simpler terms, a perfect square is the result you get when you multiply an integer by itself. Here are some examples to illustrate this concept:

• The number 25 is a perfect square because it equals \(5 \times 5\) or \(5^2\).
• Similarly, 9 is a perfect square since \(3 \times 3\) or \(3^2\) equals 9.
• An example with variables would be \(x^2\), which is a perfect square because it is \(x \times x\).

Recognizing perfect squares is very useful when you're simplifying radicals. When you identify perfect squares within a radical expression, it allows you to "take out" or "break down" the radical so it's simpler or more manageable.
During this process, you separate the components inside the square root to get factors you can simplify – for instance, transforming \(\sqrt{50}\) into \(\sqrt{25} \times \sqrt{2}\). This transformation works because 25 is a perfect square.

Radical Expressions

Radical expressions include any mathematical term involving a radical symbol (\(\sqrt{}\)). The most common type you will encounter is the square root, but there are many others with roots like cube roots or fourth roots. Let's dive deeper into their structure and simplification.

• The radical symbol \(\sqrt{}\) itself signifies that you are dealing with square roots, unless indicated otherwise.
• Within the radical, you will often have numbers (or expression) that can be broken down into parts, as seen with \(\sqrt{50}\).
• The goal is to simplify these expressions by factoring out perfect squares.

In our example, transforming \(\sqrt{50x^2}\) into a simpler form involves recognizing that 50 can be factored into 25 and 2. This breakdown allows further simplification because \(\sqrt{25}\) is a whole number, specifically 5. The variable part, \(x^2\), can also be simplified as \(\sqrt{x^2} = x\), because it too is a perfect square.
After simplification, the original expression \(\sqrt{50x^2}\) becomes \(5x\sqrt{2}\).

Square Roots

A square root of a number \(a\) is another number that, when multiplied by itself, gives \(a\). The square root is indicated by the radical sign \(\sqrt{}\), with the number or expression underneath it being referred to as the radicand.

• The square root of 25 is 5, because \(5 \times 5 = 25\).
• Generally speaking, the square root of any perfect square \(n^2\) is \(n\).
• When working with variables, the square root of any variable squared, such as \(x^2\), is simply \(x\).

Square roots are especially important when simplifying radical expressions. Knowing them allows you to break down complex radicals into more basic components. You should always search for perfect squares that can be "taken out" of the radical.
In the expression \(5x\sqrt{2}\), we've taken 5 out as the square root of 25, and \(x\) out as the square root of \(x^2\), making the expression much easier to work with and understand.