Problem 10 - College Algebra Essentials

Question

Evaluate each expression in Exercises \(1-12,\) or indicate that the root is not a real number. $$\sqrt{144}+\sqrt{25}$$

Step-by-Step Solution

Answer

The evaluated result of the expression \(\sqrt{144} + \sqrt{25}\) is 17.

Step 1: Calculate the square root of the first number

Starting with the first number inside the square root sign \(\sqrt{144}\), the square root of 144 is calculated. Since 12*12 = 144, the square root of 144 is 12.

Step 2: Calculate the square root of the second number

Moving to the second number inside the square root sign \(\sqrt{25}\), the square root of 25 is calculated. Since 5*5 = 25, the square root of 25 is 5.

Step 3: Perform the Addition

Lastly, add the two results found in the previous steps together. Which becomes 12 + 5 = 17.

Key Concepts

Real Numbers

Real numbers are fundamental in mathematics and are vital for understanding concepts such as square roots. They include all the numbers we typically work with in daily life. If you can find it on the number line, it's a real number. This set includes:

• Rational numbers: Numbers that can be expressed as a fraction, like 1/2 or 3.25.
• Irrational numbers: Numbers that can't be written as a simple fraction, such as \(\pi\) or \(\sqrt{2}\).
• Whole numbers: Non-negative numbers without fractions or decimals, such as 0, 1, 2, and so on.

Square roots often result in real numbers, provided the number inside the square root is non-negative. For instance, \(\sqrt{144}\) results in the real number 12. Understanding this helps in evaluating expressions involving square roots, as it assures that results like \(\sqrt{25}\) are real, tangible numbers valuable in all kinds of calculations.

Addition of Square Roots

Adding square roots can be straightforward when each root results in a non-decimal real number. For expression \(\sqrt{144} + \sqrt{25}\), you start by evaluating each square root separately.

• Step 1: First, find the square root of 144, which is 12. This is because 12 times 12 equals 144.
• Step 2: Next, compute the square root of 25, arriving at 5 since 5 times 5 gives 25.
• Step 3: Lastly, sum these two results. The addition of 12 (from \(\sqrt{144}\)) and 5 (from \(\sqrt{25}\)) gives 17.

The key to adding square roots is first finding the actual real numbers they represent. If each root results in a real number, adding them is as simple as regular addition. However, if the square root does not yield a real number, because of a negative under the square root sign, the sum would not fall into our familiar realm of real numbers.

Evaluating Expressions

Evaluating expressions involves breaking down the expression into simpler parts to solve it. With square roots, it specifically entails simplifying the root first.

• Start by identifying each number under the root and calculating its square root. For instance, \(\sqrt{144}\ = 12\).
• Perform any operations laid out in the expression, such as addition, as in \(\sqrt{144} + \sqrt{25} = 12 + 5 = 17\).
• Ensure each part of the expression leads to a real and legitimate result, confirming that no part defies mathematical laws, like the square root of a negative number in the real number system.

This skill becomes particularly handy as it ensures a clear, structured approach to solving mathematical problems, where each part of the expression contributes accurately to the final answer.