Problem 6 - College Algebra

Question

Evaluate each expression in Exercises \(1-6\) or indicate that the root is not a real number. $$\sqrt{(-17)^{2}}$$

Step-by-Step Solution

Answer

The evaluated expression is 17

Step 1: Calculate the Square of -17

The first step involves squaring the number inside the brackets, which is -17. To calculate this, multiply -17 by itself: \((-17)^2 = 289\)

Step 2: Take the Square Root of 289

Next step involves taking the square root of 289, which results in a non-negative number: \(\sqrt{289} = 17\)

Key Concepts

Square Numbers

To fully grasp the concept of square numbers, it's important to go back to the basics. Square numbers, often called perfect squares, are the result of an integer multiplied by itself. For instance, if you take the number 5 and multiply it by itself (5 x 5), you get 25, which is a square number. In mathematical terms, if an integer is represented by 'n', then its square is written as \( n^2 \).

These numbers are foundational in many areas of mathematics, including geometry where they represent the area of a square with sides of equal length. They also play a crucial role in the evaluation of square root expressions. When you take the square root of a square number, you'll get the original number back. For example, \( \sqrt{25} = 5 \). This is why, in our original exercise, the square root of \( 289 \) is \( 17 \), because \( 17^2 = 289 \).

Radical Expressions

Moving on to radical expressions, let's understand what they are and how they function. A radical expression is any mathematical expression containing a radical symbol (\( \sqrt{} \)) with a number inside. The most typical radical expression is the square root, but radical expressions can also represent cube roots, fourth roots, and so on.

When evaluating a square root, such as \( \sqrt{289} \), we are searching for a number which, when multiplied by itself, gives the number under the radical. It's important to remember that square roots will always yield a non-negative result, even if the number under the radical was obtained by squaring a negative number, as seen in our exercise with the negative number -17. If a square root does not result in a rational number, it's called an 'irrational' number, such as \( \sqrt{2} \). In our exercise, since \( 289 \) is a perfect square, its square root is simply \( 17 \).

Complex Numbers

Lastly, while not directly applicable to the exercise given, it's valuable to be aware of complex numbers, especially in context with square roots of negative numbers. A complex number is a number that can be expressed in the form \( a + bi \), where 'a' and 'b' are real numbers, and 'i' is the imaginary unit, satisfying the equation \( i^2 = -1 \). We encounter complex numbers when taking the square root of a negative number, which is not possible within the set of real numbers.

For example, \( \sqrt{-1} \) is not a real number—it's actually the imaginary unit 'i'. If we were asked to find the square root of \( -289 \) instead, the result would be \( 17i \), indicating it lies in the complex plane, not on the real number line. The concept of complex numbers expands our understanding beyond the limitation that square roots have to be non-negative and opens up a whole new field of mathematics called complex analysis.