Problem 14 - Algebra A Combined Function

Question

Find each square root. Assume that all variables represent nonnegative real numbers. $$ -\sqrt{9} $$

Step-by-Step Solution

Answer

The expression \(-\sqrt{9}\) evaluates to \(-3\).

Step 1: Identify the Square Root Expression

The expression given is \(-\sqrt{9}\). This means we are looking for the negative of the square root of 9.

Step 2: Calculate the Square Root

Find the square root of 9. Since 9 is a perfect square, its square root is 3. Therefore, \(\sqrt{9} = 3\).

Step 3: Apply the Negative Sign

Now apply the negative sign to the square root we found. So, \(-\sqrt{9} = -3\).

Key Concepts

Perfect Squares

Perfect squares are numbers that can be expressed as the product of an integer with itself. For example, when we multiply 3 by itself, we obtain 9. Hence, 9 is a perfect square. Similarly, numbers like 1, 4, 16, and 25 are also perfect squares because they can be expressed as \(1^2\), \(2^2\), \(4^2\), and \(5^2\) respectively. Identifying perfect squares is a useful skill because their square roots will always be whole numbers. This simplifies calculations immensely.

It's important to remember that perfect squares are non-negative. This is because squaring any integer, whether positive, negative, or zero, will result in a non-negative number. However, the square roots of perfect squares can be both positive and negative since squaring a positive or negative number gives the same perfect square.

Nonnegative Real Numbers

Nonnegative real numbers are numbers that are greater than or equal to zero. They include all positive numbers and zero, but exclude negative numbers. This concept is crucial when dealing with square roots because the operation of taking the square root is often defined only for nonnegative numbers within the real number system.

For instance, when you see the term \(\sqrt{9}\), it assumes the number under the square root is nonnegative, leading to a result of 3, which itself is a nonnegative number. This assumption holds for all real numbers; thus, a square root expression involving negative numbers would be invalid or complex when dealing with real numbers. Moreover, this property ensures we end up with results that can be handled easily within real number arithmetic without involving imaginary units.

Negative Sign in Square Roots

The presence of a negative sign in front of a square root, such as in the expression \(-\sqrt{9}\), signifies that you should first find the square root and then apply the negative sign to that result. It’s important to understand that the negative sign is not a part of the square root operation itself but is applied independently after calculating the square root.

For example, we first determine \(\sqrt{9} = 3\) as 9 is a perfect square. Then, by applying the negative sign in front, we conclude \(-\sqrt{9} = -3\). This procedure is essential to correctly solving problems with negative square roots. Just remember that the negative indicates that you take the opposite of the positive square root, hence resulting in a negative number if the original number was positive.