Problem 37 - College Algebra with Modeling and Visualization

Question

Simplify the radical expression by factoring out the largest perfect nth power. Assume that all variables are positive. $$ \sqrt{200} $$

Step-by-Step Solution

Answer

The simplified form of \(\sqrt{200}\) is \(10\sqrt{2}\).

Step 1: Identify Perfect Squares

List all perfect squares that are factors of 200: 1, 4, 9, 16, 25, 36, etc. Not all are factors of 200, but notice that 4 and 25 are. In fact, 4 and 25 are the perfect squares that multiply to give 100, so the largest perfect square is 100.

Step 2: Express 200 as a Product of Perfect Squares

Write 200 as a product of its largest perfect square factor. Since 100 is a perfect square and \(200 = 100 \times 2\), we can express 200 as same, \(200 = 10^2 \times 2\).

Step 3: Simplify the Square Root

Take the square root of the perfect square and multiply it by the square root of the remaining factor. \(\sqrt{200} = \sqrt{100 \times 2} = \sqrt{100} \times \sqrt{2} = 10 \sqrt{2}\). Thus, the simplified form is \(10 \sqrt{2}\).

Key Concepts

Simplifying Radicals

When working with radical expressions, one of the common tasks is simplifying them, which means rewriting the expression in its simplest form. Simplifying radicals involves identifying and extracting perfect square factors from inside the radical sign.

• First, recognize that factors of a number can include perfect squares. Perfect squares are numbers like 4, 9, 16, 25, etc., which are squares of integers.
• The goal is to find the largest perfect square factor of the number inside the radical. By doing so, you can "simplify" the expression by removing the square root of that factor from the radical.

To simplify, you'll express the number as a product of its perfect square factor and another number. Then, you can take the square root of the perfect square factor, leaving you with a simpler expression. In the case of \(\sqrt{200}\), the largest perfect square factor is 100, which leads to the simplified result: \(10 \sqrt{2}\).

Perfect Squares

Perfect squares play a crucial role in simplifying radicals. A perfect square is the result of squaring an integer. For example, \(1^2 = 1\), \(2^2 = 4\), \(3^2 = 9\), and so on.

• To simplify radicals, find perfect square factors of the number under the radical sign.
• Identify the largest perfect square that divides the number completely.

This concept is used to rewrite a number as a product of the largest perfect square and another factor. Perfect squares like 4 or 25, seen in numbers like 200, are factors you should look for. They allow you to take part of the radical "out of the square root," simplifying the expression significantly. For instance, since \(100\) is a perfect square and a factor of \(200\), it's used to simplify \(\sqrt{200}\) to \(10 \sqrt{2}\).

Square Roots

The operation of taking a square root is essentially the opposite of squaring a number. It's about finding a number that, when multiplied by itself, gives you the original number. The symbol \(\sqrt{}\) is used to denote the square root.

• For many numbers, like perfect squares (e.g., 4, 9, 16), the square root is an integer.
• In other cases, the square root is an irrational number, which cannot be expressed as a simple fraction.
• It's important to remember that every positive number has two square roots: a positive and a negative. In this context, we'll focus on the principal (positive) square root.

In exercises asking you to simplify square roots, the task often involves rewriting the expression to make the radical part as simple as possible. For example, \(\sqrt{200}\) is simplified to \(10 \sqrt{2}\), because \(\sqrt{100}\) is \(10\), and the expression breaks down into simpler parts for easier manipulation or further calculations.