Problem 19 - Elementary Algebra

Question

Simplify each expression by performing the indicated operation. $$ \sqrt{200}-\sqrt{128} $$

Step-by-Step Solution

Answer

Question: Simplify the expression $\sqrt{200} - \sqrt{128}$. Answer: $2\sqrt{2}$

Step 1: Simplify the square roots

First, we will simplify the two square roots in the expression separately. To simplify a square root, we need to look for the largest perfect square that is a factor of the number under the square root. Let's start with the first square root: $$ \sqrt{200} $$ We can observe that \(200 = 100 \times 2\). Since \(100\) is a perfect square, we can rewrite the square root as: $$ \sqrt{200} = \sqrt{100 \times 2} = \sqrt{100} \times \sqrt{2} = 10\sqrt{2} $$ Now let's do the same for the second square root: $$ \sqrt{128} $$ We find that \(128 = 64 \times 2\). Since \(64\) is a perfect square, we can rewrite the square root as: $$ \sqrt{128} = \sqrt{64 \times 2} = \sqrt{64} \times \sqrt{2} = 8\sqrt{2} $$

Step 2: Perform the subtraction

Now that we have simplified the square roots, we can go ahead and perform the subtraction: $$ \begin{aligned} 10\sqrt{2} - 8\sqrt{2} &= \left(10 - 8\right)\sqrt{2} \\ &= 2\sqrt{2} \end{aligned} $$

Step 3: Final Answer

The simplified expression is: $$ 2\sqrt{2} $$

Key Concepts

Perfect Square Factors

Understanding perfect square factors is a crucial step in simplifying radical expressions. A perfect square is a number that is the square of an integer. Examples include 1, 4, 9, 16, 25, and so on. These numbers are called perfect squares because they can be expressed as some integer multiplied by itself, like

• 4 = 2 \times 2,
• 9 = 3 \times 3,
• and 16 = 4 \times 4.

When simplifying square roots, it's helpful to identify the largest perfect square factor of a given number under the root. For instance, if we have the expression \( \sqrt{200} \), knowing that 100 is the largest perfect square factor of 200 allows us to break down the square root:

• \( \sqrt{200} = \sqrt{100 \times 2} = \sqrt{100} \times \sqrt{2} = 10\sqrt{2} \).

This simplification reduces the problem to an easier form that can be more easily manipulated, such as in operations like subtraction. Identifying the largest perfect square factor efficiently helps to transform seemingly complex expressions into manageable terms.

Simplification of Square Roots

The simplification of square roots involves breaking down a complex square root into simpler parts by using the perfect square factors. This process makes it easier to work with radical expressions, especially when performing additional operations such as addition or subtraction. To simplify a square root:

• First, find the largest perfect square that is a factor of the number inside the square root.
• Rewrite the expression as a product of two square roots—one containing the perfect square and the other containing the remaining factor.
• Simplify by taking the square root of the perfect square, which results in an integer multiplied by the remaining square root.

For example, with \( \sqrt{128} \), the process involves finding the largest perfect square factor, which is 64. We then simplify:

• \( \sqrt{128} = \sqrt{64 \times 2} = \sqrt{64} \times \sqrt{2} = 8\sqrt{2} \).

Practicing this method helps you to quickly simplify square roots and prepare expressions for further operations, such as addition or subtraction of radicals.

Subtraction of Radicals

Subtracting radicals involves a few straightforward steps, provided that the radicals have been simplified correctly. Once simplified, radicals that have the same radicand (the number under the square root) can be combined much like algebraic terms known as 'like terms'. Here's what to do:

• Simplify each radical. As seen in the exercise, for \( \sqrt{200} \) and \( \sqrt{128} \) we have simplified these to \( 10\sqrt{2} \) and \( 8\sqrt{2} \), respectively.
• Subtract the coefficients of the like terms while maintaining the common radical part.
• So, \( 10\sqrt{2} - 8\sqrt{2} \) simplifies to \( (10-8)\sqrt{2} = 2\sqrt{2} \).

This shows simplicity in handling radicals once they transform into like terms. Always ensure that the radicals are first simplified to their lowest form for efficient subtraction. With this approach, like terms can be added or subtracted just like any other algebraic terms, allowing for an elegant simplification of expressions involving radical subtraction.