Problem 13 - Algebra A Combined Function

Question

Find each square root. Assume that all variables represent nonnegative real numbers. $$ -\sqrt{36} $$

Step-by-Step Solution

Answer

The square root is -6.

Step 1: Identify the Expression

The expression given is \(-\sqrt{36}\). Here, the negative sign is outside the square root.

Step 2: Find the Square Root of 36

Calculate the square root of 36. Since 36 is a perfect square, we have: \(\sqrt{36} = 6\).

Step 3: Apply the Negative Sign

Attach the negative sign to the result from Step 2. So, \(-\sqrt{36} = -6\).

Step 4: Conclusion

Therefore, the solution to the problem \(-\sqrt{36}\) is \(-6\).

Key Concepts

Nonnegative Real Numbers

In mathematics, nonnegative real numbers include all the positive real numbers and zero. These numbers are vital when dealing with square roots, as the square root operation is usually defined for nonnegative numbers only. The nonnegative real numbers are represented by the set

• This set encompasses all real numbers that are greater than or equal to zero.
• It helps in avoiding complex numbers in most basic calculations related to square roots.

When given a variable or a number in an equation, if it is stated to be nonnegative, this means you can safely calculate its square root without running into issues like imaginary numbers. Often, in exercises, it's assumed that all involved variables will take on nonnegative real values.

Perfect Squares

Perfect squares are integers that are the square of an integer. Understanding perfect squares is essential when solving problems involving square roots, as they simplify calculations.

• For example, 36 is a perfect square because it is the result of squaring 6 (i.e., \(6^2 = 36\)).
• Working with perfect squares is straightforward because the square root of a perfect square is always an integer.

This feature of perfect squares allows for easy simplification in mathematical problems, making them popular in exercises and problem-solving. Recognizing patterns of perfect squares can quickly lead you to find solutions without the need for a calculator, saving time and simplifying the process.

Negative Sign in Square Roots

The negative sign in front of a square root notation can sometimes be confusing, but it has a straightforward meaning. When you see a negative sign before the square root symbol, such as \(-\sqrt{36}\), it means you take the negative of the square root's result. For example:

• First, determine the square root: \(\sqrt{36} = 6\).
• Then apply the negative sign, \(-\sqrt{36} = -6\).

This negative sign is not part of the number inside the square root. Instead, it suggests taking the standard positive square root and then multiplying it by \(-1\). Remember, the order here is important to avoid mistakes, as it can entirely change the meaning and result of your calculation.