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Understanding the properties of exponents is crucial when working with algebraic expressions. Exponents, also known as powers, represent how many times a number is multiplied by itself. For example, when we see the expression \(a^b\), this means that 'a' is being multiplied by itself 'b' times.

One of the basic properties that is particularly relevant when addressing absolute values is that any real number squared (raised to the power of 2) will always result in a non-negative number. This holds true because multiplying two positive numbers or two negative numbers together will always yield a positive result. For instance, \((-5)^2 = 25\) and \((3)^2 = 9\), both are non-negative. This property allows us to simplify expressions with absolute values, as seen in the textbook example of \(|t^2|\), where the absolute value can be removed without altering the expression.

When faced with the task of rewriting expressions, the goal is to simplify or alter the form without changing the expression's value. This is done to make the expression more understandable or easier to work with. In algebra, it's common to use rules such as the properties of exponents to rewrite expressions more efficiently.

Additionally, absolute values can sometimes be removed if we can determine that the expression within is already non-negative. The process involves examining the inner expression and assessing whether the absolute value is necessary. In the example of \(|t^2|\), since we know that the square of any real number is non-negative, we can safely drop the absolute value signs and write the expression simply as \(t^2\). This step of rewriting is important for solving more complex equations and can help simplify calculations.

Non-negative numbers include all positive real numbers and zero. In other words, they are the set of numbers that are greater than or equal to zero. These do not include any negative numbers. It is essential to distinguish between non-negative and positive numbers: while all positive numbers are non-negative, not all non-negative numbers are positive, as this category also includes zero.

The concept of non-negativity is closely related to absolute values. Absolute value refers to the distance of a number from zero on the number line, thus it is always non-negative. When assessing expressions that include absolute values, such as the textbook example \(|t^2|\), recognizing that the square of any real number is non-negative allows us to simplify expressions confidently, knowing that the original value's integrity remains intact.