Problem 20 - College Algebra

Question

rewrite each expression without absolute value bars. $$ |\sqrt{5}-13| $$

Step-by-Step Solution

Answer

hence, \(|\sqrt{5} - 13|\) can be rewritten without absolute value bars as \(13 - \sqrt{5}\).

Step 1: Check the Condition for Non-negativity

To ascertain if the expression inside the absolute value is non-negative, verify whether \(\sqrt{5} - 13 \geq 0\). This, however does not hold true because, \(\sqrt{5}\) is around 2.24 and subtracting 13 from it results in a negative number.

Step 2: Reverse the Sign of the Expression

As the result in step 1 is negative, by the nature of absolute values, we must reverse its sign. So, we have the expression as the negative of \(\sqrt{5} - 13\), which is \(13 - \sqrt{5}\).

Key Concepts

Simplifying Expressions

Simplifying expressions is an important skill in algebra that helps students to make complex algebraic expressions more manageable. The process involves combining like terms, factoring, expanding expressions, and using various algebraic properties to rewrite expressions in a simpler or more useful form. An absolute value operation, like in the exercise \( |\sqrt{5}-13| \), adds another layer to simplification, since it requires considering the non-negativity of the expression inside the bars. Simplifying an absolute value usually involves two scenarios: If the expression inside the absolute value is non-negative, it remains the same; otherwise, the expression is negated to ensure a non-negative result.

For instance, with \( |\sqrt{5}-13| \), simplification involved assessing the value inside and determining that because it was negative, it should be negated. Simplification does not alter the value of an expression—it simply makes it easier to work with.

Radicals

Radicals, or roots, are expressions that represent the inverse operation of exponentiation. The most common radical is the square root, denoted as \( \sqrt{x} \), which answers the question 'what number squared equals \(\)?'. In the given exercise, \( \sqrt{5} \), a radical, appears inside the absolute value bars.

When simplifying expressions with radicals, it's important to know that radicals can often be simplified further by factoring out perfect squares, cubes, etc., from underneath the radical. However, in our case with \( \sqrt{5} \), there are no perfect square factors, and thus the radical is already in its simplest form. In dealing with absolute values involving radicals, the key step is understanding whether the radical makes the inside expression non-negative or not.

Inequalities

Inequalities express the relationship between two values, indicating whether one is greater than, less than, equal to, or not equal to another value. They play a critical role in determining the nature of expressions within absolute value bars, as seen in our exercise. To resolve an absolute value, one must decide if the inside expression is greater than or equal to zero, which would retain its form, or less than zero, requiring a change in sign.

This approach was applied in checking whether \( \sqrt{5} - 13 \geq 0 \). After determining that the inequality was false, and the expression was indeed negative, the simplification process then involved negating the expression inside the absolute value to make it non-negative. Inequalities, therefore, serve as a decision-making tool in the process of simplifying expressions with absolute values.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and arithmetic operations, and they represent values that may change. Absolute value expressions are a subset of algebraic expressions that always yield a non-negative result, even when they contain negative inputs. The given exercise, involving \( |\sqrt{5}-13| \), features two important algebraic concepts—radicals and absolute values—illustrating the versatility of algebraic expressions.

Manipulating algebraic expressions requires an understanding of the rules for arithmetic operations, the ability to recognize the need to simplify complex expressions, and the recognition of the special cases that absolute value presents. Remember, while variables can represent any value, when simplified, as in the final form of \( 13 - \sqrt{5} \), algebraic expressions should express a clear, concise value or set of values.