Problem 15 - College Algebra with Corequisite Support

Question

For the following exercises, simplify each expression. \(\sqrt{169}+\sqrt{144}\)

Step-by-Step Solution

Answer

25

Step 1: Calculate the Square Roots

First, we calculate the square roots of each number under the radical.Start with \(\sqrt{169}\). The number 169 is a perfect square because \(13 \times 13 = 169\). Therefore, \(\sqrt{169} = 13\).Next, calculate \(\sqrt{144}\). The number 144 is a perfect square as well, since \(12 \times 12 = 144\). Thus, \(\sqrt{144} = 12\).

Step 2: Add the Results

Now that we know the square roots, add the two results together.So, we add \(13 + 12\).This gives us \(13 + 12 = 25\).

Step 3: Simplification Complete

The result \(25\) is the simplified result of the original expression \(\sqrt{169} + \sqrt{144}\). Thus, the simplification process is complete.

Key Concepts

Simplification

Simplification is the process of making something easier to work with by reducing its complexity. When dealing with square roots, simplification involves reducing the expression to its simplest form. To simplify an expression like \(\sqrt{169} + \sqrt{144}\), we first need to calculate each square root separately. By recognizing perfect squares, we can replace each radical with its corresponding whole number, resulting in an expression that is more straightforward to work with.

• Identify the perfect square factors
• Calculate the square roots
• Sum the resulting numbers

In this exercise, after calculating \(\sqrt{169}\) as 13 and \(\sqrt{144}\) as 12, the sum \(13 + 12\) equates to 25, giving us a simplified result. This step-by-step simplification not only provides the answer but also shows a streamlined approach to solving similar problems.

Perfect Squares

Perfect squares are numbers that are the product of an integer multiplied by itself. Recognizing them is essential in working with square roots, as it greatly simplifies calculations. For instance, 169 and 144 are perfect squares because:

• 169 equals \(13 \times 13\)
• 144 equals \(12 \times 12\)

Understanding perfect squares helps in immediately determining that \(\sqrt{169} = 13\) and \(\sqrt{144} = 12\). This provides a helpful shortcut in computations, making it quicker to resolve expressions containing square roots. By memorizing common perfect squares, you can enhance your ability to simplify radical expressions swiftly.

Radical Expressions

Radical expressions involve roots, most commonly square roots, represented by the radical symbol \(\sqrt{}\). The process of simplifying radical expressions often includes identifying and using perfect squares. Simplifying \(\sqrt{169} + \sqrt{144}\) involves reducing each radical to its basic integer form. Recognizing that the numbers under the radicals are perfect squares allows you to replace the radicals with integers. Thus, simplifying the expression into an easily managed sum.

• Radicals result from extracting square roots
• The primary goal is to express in the simplest form possible
• Key step is recognizing perfect squares under the radicals

This means that the original expression simplifies to merely summing these whole numbers obtained from the radicals, leading to a straightforward solution.