Problem 11 - Elementary Algebra

Question

Write the principal and secondary square roots of each number. $$ 121 $$

Step-by-Step Solution

Answer

Answer: The principal square root of 121 is 11, and the secondary square root is -11.

Step 1: Identifying the number whose square is equal to 121

To find the square root, we need to identify a number which, when multiplied by itself, gives the original number (121 in this case). We can try out some numbers here. Let's start with trying 11. $$ 11^2 = 11 \times 11 = 121 $$

Step 2: Write down the principal and secondary square roots

As we have found that 11, when multiplied by itself, results in 121, the principal square root of 121 is 11. The secondary square root is simply the negative of the principal square root. So, the secondary square root is -11. The principal and secondary square roots of 121 are: $$ \sqrt{121} = 11 $$ $$ -\sqrt{121} = -11 $$

Key Concepts

Principal Square Root

The principal square root of a number is the non-negative square root. It refers to the positive number that, when squared (multiplied by itself), equals the original value. For instance, the principal square root of 121 is 11 since multiplying 11 by itself gives 121.
\(11 \times 11 = 121\)
It is denoted with the radical symbol \(\sqrt{\cdot}\). When you see \(\sqrt{121}\), it specifically implies the non-negative root, which is 11 in this case.
If you're dealing with square roots in positive contexts, such as physics or geometry, the principal square root is what you're usually interested in. Understanding the principal square root helps you solve equations and make sense of shapes and spaces where non-negative values are required.

Secondary Square Root

The secondary square root is the negative counterpart of the principal square root. Mathematically, for any positive number, both positive and negative numbers squared will give the same positive result. So, if 11 is a square root of 121, then -11 is also a square root of 121.
When you think about the secondary square root, think of it simply as the negative version of the principal square root.
For any positive number \(a\), we denote the secondary square root as \(-\sqrt{a}\). So for 121, the secondary square root would be \(-11\), because:
\(-11 \times -11 = 121\)

• The positive \(-\times-\) equals the positive \(+\)
• The secondary square root is used in mathematics to provide more comprehensive solutions to quadratic equations.

Perfect Square

A perfect square is an integer that can be expressed as the square of another integer. This means if you multiply an integer by itself and get another integer, the result is a perfect square. For example, 121 is a perfect square because it can be written as \(11^2\).
Recognizing perfect squares is a vital part of understanding square roots since perfect squares make the process much cleaner and without decimals or fractions.

• Common perfect squares you may encounter are numbers like 1, 4, 9, 16, and 25.
• Perfect squares have whole number roots, making calculations straightforward for many math problems.

Identifying perfect squares also helps when simplifying radicals and solving algebraic equations by reducing them to simpler, more manageable forms.