Problem 43 - Elementary Algebra

Question

For the following problems, find the two square roots of the given number. $$ 1.21 $$

Step-by-Step Solution

Answer

Answer: The two square roots of 1.21 are 1.1 and -1.1.

Step 1: Find the principal square root

To find the principal square root (the positive one) of the given number, simply take the square root of 1.21. $$ \sqrt{1.21} = 1.1 $$

Step 2: Find the negative square root

To find the negative square root, simply take the negative of the principal square root, which is -1.1. $$ -\sqrt{1.21} = -1.1 $$

Step 3: Final Answer

The two square roots of 1.21 are 1.1 and -1.1.

Key Concepts

Principal Square Root

The concept of the principal square root refers to the non-negative square root of a number. When we talk about the square root of a number, without any additional description, it is generally understood to mean the principal square root. This is important because every positive real number actually has two square roots: one positive and one negative. However, the principal square root is always the positive one.

For example, to find the principal square root of 1.21, you calculate \( \sqrt{1.21} \). This gives you 1.1, which is the principal square root since it's positive. Always remember:

• Principal square root = Positive square root
• It is denoted simply by the square root symbol \( \sqrt{} \) without any signs.
• For real numbers, the principal square root is always the non-negative root.

Negative Square Root

Unlike the principal square root, the negative square root of a number is, as the name suggests, negative. To find the negative square root of a number, you simply take the principal square root and add a negative sign in front of it.

Using our previous example of 1.21, we found that the principal square root is 1.1. Then, the negative square root of 1.21 is obtained by placing a negative sign before the principal square root: \(-\sqrt{1.21} = -1.1\).

Here are some key points about the negative square root:

• It is simply the opposite of the principal square root.
• Indicated with a negative sign before the root, \(-\sqrt{}\).
• It does not indicate a new calculation but rather a change in sign only.

Positive Square Root

The positive square root is another way of referring to the principal square root, which is always positive for real numbers. This term helps to emphasize that we are considering only the non-negative root of the number when solving square root problems.

For every positive number, the positive square root is meaningful as it represents a value that, when multiplied by itself, gives the original number. For instance, the positive square root of 1.21 is 1.1, as shown in earlier sections. This follows from:

• Positive square root means the same as principal square root.
• This value is what you would typically use in mathematical contexts unless a negative solution is required.
• The symbol \( \sqrt{} \) inherently refers to the positive square root.

In summary, when you're asked for the square root of a number without further clarification, it's usually this positive or principal square root that's being requested.