Problem 12 - Algebra and Trigonometry

Question

Evaluate each expression indicate that the root is not a real number. $$ \sqrt{(-17)^{2}} $$

Step-by-Step Solution

Answer

The value of the expression \(\sqrt{(-17)^{2}}\) is 17.

Step 1: Identify the term under the square root

Our term under the square root is \((-17)^{2}\). It's important to understand that the square of a negative number (like -17) is a positive number, as negative times negative equals positive.

Step 2: Calculate the square of the number inside the brackets

To continue, we need to square -17. It translates to multiplying -17 by itself, which results in 289. So, our entire new expression becomes \(\sqrt{289}\).

Step 3: Calculate the square root

Considering that the square root is the inverse operation to squaring a number, we need to find a number which, multiplied by itself, gives 289. This number is 17. So, the square root of 289 is 17.

Key Concepts

Understanding Square Roots

A square root is essentially the opposite of squaring a number. When you square a number, you multiply it by itself. For instance, squaring 3 gives you 9. The square root works the other way around. It's about finding a number which, when multiplied by itself, returns the original number.
Imagine you have the number 289 and you want to find its square root. You are looking for a number that gives you 289 when squared. In this case, 17 is the number you need because \[17 \times 17 = 289\]
It’s important to note that while most numbers have square roots, real number square roots should result in a positive value. This positive result is called the principal square root. So, the principal square root of 289 is 17.

Working with Negative Numbers

Negative numbers are numbers less than zero. They have a "-" sign before them, such as -3 or -17.
When dealing with negative numbers, it's crucial to be aware of how operations affect them. For example, multiplying two negative numbers results in a positive number. Hence, \((-17) \times (-17) = 289\)
This is because the two minus signs cancel each other out. However, taking the square root of a negative number directly (without an intermediary operation) does not result in a real number. The square root of negative values involves another set of numbers called imaginary numbers, but when using square roots in real numbers, such cases are labeled as "not real."
In our exercise, we square -17 first, turning our focus to real numbers only.

Inverse Operations

Inverse operations are mathematical operations that undo each other. In simpler terms, they are pairs of operations that reverse the effect of each other over a set of numbers.

• Addition and subtraction are inverse operations.
• Similarly, multiplication and division are inverse operations.
• Squaring and finding a square root are also inverse operations.

In the context of the exercise, squaring is the operation of multiplying a number by itself, and finding the square root reverses this operation. In other words, when you find the square of a number and then take the square root, you end up with the original number.
For \(289\), the squaring happened when you multiplied \(-17\) by itself, resulting in \(289\). Later, finding the square root of \(289\) brings us back to the number \(17\). This demonstrates how inverse operations work to return to your starting point.