Problem 65 - College Algebra

Question

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

Step-by-Step Solution

Answer

The expression simplifies to \(\sqrt{2}\), which is irrational.

Step 1: Simplify Inside the Square Root

Let's first simplify the expression inside the square root: \[-18 - 4(5)(-1)\]Start by calculating the product inside the parentheses: \[4 \times 5 = 20\]Since it's \(4(5)(-1)\), we need to multiply the result by -1:\[20 \times (-1) = -20\]Now substitute back into the expression:\[-18 - (-20) = -18 + 20\]

Step 2: Calculate the Result of the Simplification

Now, simplify the expression \(-18 + 20\):\[-18 + 20 = 2\]So, the expression inside the square root simplifies to \(2\).

Step 3: Determine the Nature of the Result

The given expression now is \(\sqrt{2}\). Since \(2\) is not a perfect square, \(\sqrt{2}\) is not a rational number.Rational numbers are those that can be expressed as the quotient of two integers. The square root of a non-perfect square is an irrational number.

Key Concepts

Square Root Simplification

Simplifying square roots is a fundamental aspect of solving equations, especially when working with rational and irrational numbers. The challenge is to reduce the expression within the square root to its simplest form.
This involves applying arithmetic operations inside the square root and breaking the expression into smaller parts if necessary. For instance, if you encounter an expression like \( \sqrt{-18-4(5)(-1)} \), you will need to evaluate the arithmetic inside the square root first and simplify it step by step. This involves calculating the integers' product first and then following through with any arithmetic operations such as subtraction or addition.

• First, resolve any parentheses in the expression.
• Next, perform multiplication or division operations.
• Finally, add or subtract the results to simplify the expression within the square root.

A simplified expression helps in determining whether the result inside the square root leads to a rational or irrational number.

Perfect Square

Understanding the concept of a perfect square is vital when dealing with square roots. A perfect square is an integer that is the square of another integer. For example, numbers like 1, 4, 9, 16, and 25 are perfect squares because they are equal to \( 1^2 \), \( 2^2 \), \( 3^2 \), \( 4^2 \), and \( 5^2 \) respectively.
If the number inside the square root is a perfect square, the square root of that number is rational.

• For example, \( \sqrt{16} = 4 \) because 16 is a perfect square.
• If it's not a perfect square, the square root results in an irrational number.

In the problem case presented, the simplified expression inside the square root is \( 2 \), which is not a perfect square. Hence, \( \sqrt{2} \) results in an irrational number. Recognizing perfect squares immediately prompts you to foresee the nature of the square root, making a significant difference in solving problems efficiently.

Product of Integers

When simplifying expressions, particularly those involving parentheses, understanding how to handle the product of integers is critical. The product of integers refers to multiplying two or more integers together, following the rules dictated by basic arithmetic.

• Multiplying positives: \( 4 \times 5 = 20 \)
• Incorporating a negative number: \( 20 \times (-1) = -20 \)

The interplay between positive and negative numbers is pivotal. Multiplication involving negative numbers changes the sign of the product, as observed in the example \( 4(5)(-1) \). Here, while \( 4 \times 5 \) gives a positive 20, the subsequent multiplication by \(-1\) yields a negative \(-20\). Mastering these calculations is a small but vital part of larger problem-solving processes, helping to efficiently manage expressions and equations.

Quotient of Two Integers

A rational number is defined as a number that can be expressed as the quotient of two integers. This means the number should be able to be represented as \( \frac{a}{b} \) where \( a \) and \( b \) are integers and \( b eq 0 \). Rational numbers include fractions, integers, and finite or recurring decimals.

• Examples: \( \frac{1}{2}, -3, \frac{7}{1} \)

In contrast, irrational numbers cannot be written in the form of a simple fraction. Numbers such as \( \sqrt{2} \) and \( \pi \) are irrational because they result in non-repeating, non-terminating decimals.In the context of the given problem, after simplifying the expression to \( \sqrt{2} \), we find that \( \sqrt{2} \) cannot be expressed as \( \frac{a}{b} \), thus identifying it as an irrational number.