Problem 21 - College Algebra with Modeling and Visualization

Question

Evaluate by hand. $$ \sqrt{9-5}-\frac{8-4}{4-2} $$

Step-by-Step Solution

Answer

The expression evaluates to 0.

Step 1: Simplify the Expression Under the Square Root

Begin by simplifying the expression inside the square root, which is \(9 - 5\). Calculate this: \(9 - 5 = 4\), so the expression becomes \(\sqrt{4}\).

Step 2: Evaluate the Square Root

Now, calculate the square root of 4. We know that \(\sqrt{4} = 2\). So, the expression simplifies to \(2\).

Step 3: Simplify the Numerator of the Fraction

Next, simplify the numerator of the fraction, which is \(8 - 4\). Calculate this: \(8 - 4 = 4\).

Step 4: Simplify the Denominator of the Fraction

Simplify the denominator of the fraction, which is \(4 - 2\). Calculate this: \(4 - 2 = 2\).

Step 5: Evaluate the Fraction

Now, evaluate the fraction \(\frac{4}{2}\). We calculate \(\frac{4}{2} = 2\).

Step 6: Final Calculation

Finally, subtract the result of the fraction from the square root result: \(2 - 2 = 0\). The full expression evaluates to 0.

Key Concepts

Simplifying Expressions

Simplifying expressions helps break down complex problems into smaller, more manageable parts. By performing arithmetic operations and reducing expressions, we make it easier to work with them. For example, in the given exercise, by simplifying the expression inside the square root \(9 - 5\), we calculate \(9 - 5 = 4\). This reduces the task and makes it more straightforward.
Let's briefly explore what goes into simplifying an expression:

• Identify parts of the expression that can be simplified (like terms, arithmetic inside brackets, etc.)
• Perform arithmetic operations where possible (addition, subtraction, multiplication, division)
• Keep track of changes to ensure each step keeps the expression equivalent to its original form

Remember, simplifying is about making the expression as neat as possible without changing its value.

Square Roots

The square root operation is an important mathematical concept used to determine the original value when squared gives the number inside the root. For instance, \(\sqrt{4}\) asks what number, when multiplied by itself, results in 4.
The answer is 2, since \(2 \times 2 = 4\). In our exercise, simplifying \(\sqrt{9-5}\) to \(\sqrt{4}\) and then calculating \(\sqrt{4} = 2\) simplifies the expression significantly.
A few essential points about square roots include:

• A square root of any positive number usually has a positive and a negative answer, though often we only consider the positive one for simplicity.
• The square root of zero is zero, as \(0 \times 0 = 0\).
• Numbers that do not have a perfect square root can often be left in square root form for precision.

Understanding square roots helps in many areas of mathematics from geometry to algebra.

Fractions

Fractions represent parts of a whole and consist of a numerator and a denominator. In this problem, the fraction is \(\frac{8-4}{4-2}\). Simplifying each part step by step makes calculation easier.
First, simplify the numerator: \(8 - 4 = 4\). Then simplify the denominator: \(4 - 2 = 2\). Now, you can evaluate the fraction: \(\frac{4}{2} = 2\).
Fractions can represent division, which allows us to precisely tackle portions of numbers. Here are some helpful notes on working with fractions:

• Always simplify fractions to their lowest terms for ease of interpretation and calculation.
• Remember that fractions are just division problems waiting to be solved.
• Be mindful of the rules such as same denominators for adding or subtracting fractions.

Simplified fractions help avoid errors and ease reading and comprehension.

Arithmetic Operations

Arithmetic operations are at the core of evaluating any expression. The basic operations are addition, subtraction, multiplication, and division. These operations are the building blocks for all other mathematical processes.
In our given problem, subtraction and division are prominently featured in both simplifying steps and fraction evaluation. Subtraction is used in simplifying terms, and division is observed when dealing with the fraction \(\frac{4}{2}\) resulting in 2.
Understanding arithmetic operations is crucial because they:

• Assist in breaking down complex expressions into manageable steps.
• Form the basis for more advanced mathematical concepts.
• Allow for precise calculation and manipulation of numbers in various forms.

Mastery of these operations enables you to solve a wide array of math problems with confidence.