Problem 10 - Algebra and Trigonometry

Question

Evaluate each expression indicate that the root is not a real number. $$ \sqrt{144}+\sqrt{25} $$

Step-by-Step Solution

Answer

The sum of \( \sqrt{144} \) and \( \sqrt{25} \) is 17.

Step 1: Identify and Understand the Problem

Here, we are asked to find the root of 144 and 25. A root is a value that, when multiplied by itself, will provide the initial number. We use the square root symbol to represent it.

Step 2: Calculate the Square Root of 144

The square root of 144 is a number which, when multiplied by itself, will provide 144. Here, the square root of 144, denoted as \( \sqrt{144} \) is 12. Because \(12 * 12 = 144\).

Step 3: Calculate the Square Root of 25

The square root of 25 is a number which, when multiplied by itself, will provide 25. So, the square root of 25, denoted as \( \sqrt{25} \) is 5. Because \(5 * 5 = 25\).

Step 4: Add the Square Roots

The addition of \( \sqrt{144} \) and \( \sqrt{25} \) will be \(12 + 5 = 17\).

Key Concepts

Adding Square Roots

When adding square roots, there are a few key points to remember. First, you need to evaluate each square root separately before performing the addition. In our original exercise, we calculated the square roots individually:

• The square root of 144 is 12 because 12 times 12 equals 144.
• The square root of 25 is 5 because 5 times 5 equals 25.

After evaluating these, we add them together by simply collecting the numbers, just like normal addition. In this case, 12 plus 5 equals 17. This step is crucial because it confirms that we end up with a real number result when adding square roots.

Evaluating Expressions

Evaluating expressions involves more than just calculating numbers. It requires understanding how each mathematical operation works and how to apply them in the right sequence. In our exercise, we were given a simple expression to evaluate:

• First, calculate each square root to transform the expression from one with surds into basic numbers. For example, turning \( \sqrt{144} \) into \( 12 \).
• After evaluating each component, we then perform the operations listed, which is adding in this case.

It's essential to follow the order of operations, although our problem did not involve complex operations or multiple steps. Evaluating an expression can become intricate in more complex equations, so practicing such skills on simple problems helps build a foundational understanding.

Identifying Real and Non-Real Numbers

Square roots are often used to determine if a number is real or not. Understanding the nature of square roots helps in identifying real and non-real numbers in expressions.

• A square root results in a real number if the original number (under the square root) is positive and there is an exact square root.
• For example, from our exercise, both 144 and 25 are positive integers, and they each had clear square roots (12 and 5).
• If a negative number is placed under the square root in real number arithmetic, it leads to a non-real (imaginary) number. For example, \( \sqrt{-1} \) is not a real number.

Being able to identify when a square root produces a real versus a non-real number is a critical skill in math. This understanding ensures students can correctly evaluate and interpret mathematical expressions containing square roots.