Problem 65 - College Algebra

Question

Determine whether the simplified expression is rational or irrational: \(\sqrt{-18-4(5)(-1)}\)

Step-by-Step Solution

Answer

The simplified expression is irrational.

Step 1: Simplify the Expression inside the Square Root

Let's simplify the expression inside the square root: \(-18 - 4(5)(-1)\). This can be broken down as follows:Calculate \(4 \times 5 = 20\).Then multiply by \(-1\) to get \(-20\).So, the expression inside the square root becomes \(-18 - (-20) = -18 + 20 = 2\).

Step 2: Evaluate the Square Root

Now we evaluate the square root:The expression inside the square root simplifies to \(\sqrt{2}\).\(\sqrt{2}\) cannot be expressed as a fraction of two integers, making it an irrational number.

Step 3: Determine Rationality

Since the result of the expression is \(\sqrt{2}\), and we know that \(\sqrt{2}\) is an irrational number, the simplified expression is irrational.

Key Concepts

Simplifying Expressions

Simplifying expressions is a fundamental skill in mathematics. It involves breaking down complex mathematical problems into simpler, more manageable parts.
In the given exercise, the expression inside the square root is \[-18 - 4(5)(-1)\]. You start by performing the operations step-by-step:

• First, multiply 4 by 5, which equals 20.
• Next, multiply 20 by -1, resulting in -20.
• Finally, simplify the expression: \[-18 - (-20) = -18 + 20 = 2.\]

By simplifying expressions, you reduce the chance of making mistakes during calculations and create solutions that are easier to understand and work with. It is always important to follow the correct order of operations, which is often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
This ensures each part of the expression is handled accurately.

Square Roots

Square roots are a special kind of radical expression and are important in both arithmetic and algebra. The square root of a number \( x \), written as \( \sqrt{x} \), is a value that, when multiplied by itself, equals \( x \). For example, the square root of 4, which is \( \sqrt{4} \), is 2 because \( 2 \times 2 = 4 \).
In this exercise, after simplifying the expression inside the square root, we are left with \( \sqrt{2} \).

• Unlike \( \sqrt{4} \), which is a whole number, \( \sqrt{2} \) cannot be perfectly expressed as a simple fraction or rational number.
• This means \( \sqrt{2} \) is an irrational number.

Irrational numbers have non-repeating, non-terminating decimal expansions and cannot be written as a ratio of two integers. Understanding square roots and their properties helps in assessing whether an expression is rational or irrational.

Evaluating Expressions

Evaluating mathematical expressions involves finding their numerical value. This is often the last step in solving problems and requires applying previously learned concepts, like simplification and square roots.
The given problem tasked us with determining the nature of \( \sqrt{2} \). After simplifying the expression inside the root and calculating the square root, we find the value was not a perfect square.

• Specifically, the final step involves interpreting \( \sqrt{2} \).
• Since it cannot be expressed as a ratio of integers, it is determined to be irrational.

This type of evaluation often requires tapping into properties of numbers, especially the understanding of rational and irrational numbers. Evaluating expressions accurately affects not just this specific problem, but forms a basis for tackling more complex mathematical challenges.