91Ó°ÊÓ

When we talk about the square of a number, we are essentially referring to that number multiplied by itself. It's an operation most often denoted as raising the number to the power of two.
Simply put, if we have a number, let's call it 'a', then the square of 'a' is written as \( a^2 \), which is equal to \( a \times a \). For example, the square of 4 is \( 4^2 = 4 \times 4 = 16 \).
When squaring a negative number, the result is always positive, because multiplying two negative numbers gives a positive product. So, if we square -13, we calculate \( (-13)^2 = (-13) \times (-13) = 169 \), as seen in our exercise solution. This is an integral concept in algebra, which forms the basis for various equations and functions.

The root of a real number can be considered the opposite operation of taking the square. While squaring a number gives us a value that is produced by multiplying the number by itself, taking the square root gives us a value that, when squared, returns the original number.
The square root is denoted by the symbol \( \sqrt{ \ } \), and finding the square root of a number is like asking, 'What number multiplied by itself gives me this number?' So when we see \( \sqrt{169} \), we look for a number which, when multiplied by itself, gives 169. In this case, the answer is 13 because \( 13 \times 13 = 169 \).
It's important to remember that every positive real number has two square roots - one positive and one negative. However, in most elementary contexts, when we refer to 'the square root,' we are typically talking about the positive root.

Simplifying square roots involves reducing a square root to its simplest radical form. This process usually involves removing perfect squares from under the radical or breaking down the number into its prime factors and pairing them off.
For example, to simplify \( \sqrt{50} \), we would look for perfect square factors of 50. Since 25 is a perfect square and a factor of 50, we can break it down as \( \sqrt{25 \times 2} \) which simplifies to \( 5\sqrt{2} \). When we simplify square roots, the goal is to make the expression easier to understand and work with, particularly when it's part of a larger equation or function.
In the context of our textbook exercise, since 169 is a perfect square (13 times 13), the square root of 169 simplifies directly to 13 without the need for further reduction. It is already in its simplest radical form.