Problem 81 - College Algebra and Calculus: An Applied Approach

Question

Calculator Write the keystrokes you can use to evaluate \(\frac{4-\sqrt{7}}{3}\) in one step on your calculator.

Step-by-Step Solution

Answer

To solve the equation \(\frac{4-\sqrt{7}}{3}\) in one step on your calculator, input (4 - sqrt(7))/3 and press =.

Step 1: Input the numerator

In order to perform the calculations in one step on your calculator, you must understand the order of operations. Begin by extending the brackets to include the entire numerator. So on your calculator, you input (4 - \sqrt{7}) to cover the numerator of the fraction.

Step 2: Input the denominator

After typing the numerator as (4 - \sqrt{7}), immediately continue inputting the denominator by typing /3. This signifies the division by 3 which is the denominator of the fraction.

Step 3: Evaluate the expression

After the expression is completely typed into the calculator as (4 - \sqrt{7})/3, press the equals button (=) to solve for the final value.

Key Concepts

Order of Operations

Understanding the order of operations is like following a recipe. You must do things in the right sequence to get the desired result. In math, this means performing calculations in the correct order to ensure accuracy. The commonly used acronym to help remember this order is PEMDAS:

• Parentheses
• Exponents
• Multiplication and Division (left to right)
• Addition and Subtraction (left to right)

When evaluating the expression \((4 - \sqrt{7})/3\), we first handle what's inside the parentheses: \(4 - \sqrt{7}\). By solving inside the parentheses first, and then moving to division, you correctly apply the order of operations which prevents errors in your calculations. This order ensures that every component of the calculation is considered in its proper sequence, yielding an accurate result.

Evaluating Expressions

Evaluating expressions involves calculating the value of an algebraic expression by performing the operations in the correct order. In our example, \((4 - \sqrt{7})/3\), each part of the expression has a role to play.
To evaluate this expression:

• Start with the operation inside the parentheses: \(4 - \sqrt{7}\). This means you subtract the square root of 7 from 4.
• Then, proceed to simplify the expression by dividing the result by 3, as indicated by the division sign.

By inputting the entire sequence on your calculator in one go, you ensure that each operation is executed in a timely manner. Using the calculator helps manage more complex operations and reduces chances of manual error, but understanding the expression's mechanics is key to knowing what the calculator is doing behind the scenes.

Using Square Roots

Square roots are used to find a number which, when multiplied by itself, gives the original number. For instance, the square root of 9 is 3, because 3 times 3 equals 9. The expression \(\sqrt{7}\) is an irrational number, meaning it can’t be expressed as a simple fraction and has a non-repeating, non-terminating decimal.
When using a calculator to compute expressions involving square roots, you often have a specific key or function denoted by \(\sqrt{}\).
To evaluate a square root on your calculator, you might:

• Press the \(\sqrt{}\) button
• Enter the number you want to find the square root of (e.g., 7), and then close the parentheses if your calculator requires it.

Calculators can handle the complexities of irrationals effortlessly, which is useful when these numbers are part of larger calculations, like dividing them as in \((4 - \sqrt{7})/3\). Learning to use the square root function properly is essential for handling many math problems efficiently.